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A HISTORY OF
MATHEMATICAL NOTATIONS
VOLUME I
NOTATIONS IN ELEMENTARY
MATHEMATICS
A HISTORY OF
ATHEMATICAL
NOTATIONS
BY
FLORIAN CAJORJ^H.D.
Professor of the History of Mathematics
University of California
VOLUME 1
NOTATIONS IN ELEMENTARY
MATHEMATICS
THE OPEN COURT COMPANY.
PUBLISHERS,
86, STRAND, LONDON, W.C.2.
COPYRIGHT 1928 BY
THE OPEN COURT PUBLISHING COMPANY
Published September 1928
Composed and Printed By
The University of Chicago Preu
Chicago. Illinois. U.S.A.
PREFACE
The study of the history of mathematical notations was sug-
gested to me by Professor E. H. Moore, of the University of Chicago.
To him and to Professor M. W. Haskell, of the University of California,
I am indebted for encouragement in the pursuit of this research. As
completed in August, 1925, the present history was intended to be
brought out in one volume. To Professor H. E. Slaught, of the Uni-
versity of Chicago, I owe the suggestion that the work be divided into
two volumes, of which the first should limit itself to the history of
symbols in elementary mathematics, since such a volume would ap-
peal to a wider constituency of readers than would be the case with
the part on symbols in higher mathematics. To Professor Slaught I
also owe generous and vital assistance in many other ways. He exam-
ined the entire manuscript of this work in detail, and brought it to
the sympathetic attention of the Open Court Publishing Company. I
desire to record my gratitude to Mrs. Mary Hegeler Carus, president
of the Open Court Publishing Company, for undertaking this expen-
sive publication from which no financial profits can be expected to
accrue.
I gratefully acknowledge the assistance in the reading of the proofs
of part of this history rendered by Professor Haskell, of the Uni-
versity of California; Professor R. C. Archibald, of Brown University;
and Professor L. C. Karpinski, of the University of Michigan.
FLORIAN CAJORI
UNIVERSITY OF CALIFORNIA
. TABLE OF CONTENTS
I. INTRODUCTION
PARAGRAPHS
II. NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS . . . 1-99
Babylonians 1-15
Egyptians 16-26
Phoenicians and Syrians 27-28
Hebrews 29-31
Greeks 32-44
Early Arabs 45
Romans 46-61
Peruvian and North American Knot Records .... 62-65
Aztecs 66-67
Maya 68
Chinese and Japanese 69-73
Hindu- Arabic Numerals 74-99
Introduction 74-77
Principle of Local Value 78-80
Forms of Numerals 81-88
Freak Forms 89
Negative Numerals 90
Grouping of Digits in Numeration 91
The Spanish Calderon 92-93
The Portuguese Cifrao 94
Relative Size of Numerals in Tables 95
Fanciful Hypotheses on the Origin of Numeral Forms . 96
A Sporadic Artificial System 97
General Remarks 98
Opinion of Laplace 99
III. SYMBOLS IN ARITHMETIC AND ALGEBRA (ELEMENTARY PART) 100
A. Groups of Symbols Used by Individual Writers ... 101
Greeks — Diophantus, Third Century A.D 101-5
Hindu — Brahmagupta, Seventh Century .... 106-8
Hindu — The Bakhshal! Manuscript 109
Hindu— Bhaskara, Twelfth Century 110-14
Arabic — al-Khow&rizmi, Ninth Century .... 115
Arabic — al-Karkhf, Eleventh Century 116
Byzantine — Michael Psellus, Eleventh Century . . 117
Arabic — Ibn Albanna, Thirteenth Century ... 118
Chinese— Chu Shih-Chieh, Fourteenth Century . .119, 120
vii
viii TABLE OF CONTENTS
PARAGRAPHS
Byzantine — Maximus Planudes, Fourteenth Century 121
Italian — Leonardo of Pisa, Thirteenth Century . . 122
French — Nicole Oresme, Fourteenth Century . . . 123
Arabic— al-Qalasadi, Fifteenth Century .... 124
German — Regiomontanus, Fifteenth Century . . . 125-27
Italian—Earliest Printed Arithmetic, 1478 . . . . 128
French— Nicolas Chuquet, 1484 129-31
French— Estienne de la Roche, 1520 132
Italian— Pietro Borgi, 1484, 1488 133
Italian— Luca Pacioli, 1494, 1523 134-38
Italian— F. Ghaligai, 1521, 1548, 1552 139
Italian— H. Cardan, 1532, 1545, 1570 140, 141
Italian— Nicolo Tartaglia, 1506-60 142, 143
Italian— Rafaele Bombelli, 1572 144, 145
German— Johann Widman, 1489, 1526 146
Austrian — Grarnrnateus, 1518, 1535 147
German— Christoff Rudolff, 1525 148, 149
Dutch — Gielis van der Hoecke, 1537 150
German— Michael Stifel, 1544, 1545, 1553 .... 151-56
German — Nicolaus Copernicus, 1566 157
German— Johann Scheubel, 1545, 1551 .... 158, 159
Maltese— Wil. Klebitius, 1565 160
German — Christophorus Clavius, 1608 161
Belgium— Simon Stevin, 1585 162, 163
Lorraine— Albert Girard, 1629 164
German-Spanish—Marco Aurel, 1552 165
Portuguese-Spanish — Pedro Nunez, 1567 .... 166
English— Robert Recorde, 1543(?), 1557 .... 167-68
English— John Dee, 1570 169
English — Leonard and Thomas Digges, 1579 . . . 170
English— Thomas Mastcrson, 1592 171
French — Jacques Peletier, 1554 172
French— Jean Buteon, 1559 173
French — Guillaume Gosselin, 1577 174
French— Francis Vieta, 1591 176-78
Italian — Bonaventura Cavalieri, 1647 179
English— William Oughtred, 1631, 1632, 1657 . . . 180-87
English— Thomas Harriot, 1631 188
French— Pierre HSrigone, 1634, 1644 189
Scot-French—James Hume, 1635, 1636 .... 190
French — Rene* Descartes 191
English — Isaac Barrow 192
English— Richard Rawlinson, 1655-68 193
Swiss — Johann Heinrich Rahn 194
TABLE OF CONTENTS ix
PARAGRAPHS
English— John Wallis, 1655, 1657, 1685 .... 195, 196
Extract from Ada eruditorum, Leipzig, 1708 . . . 197
Extract from Miscellanea Berolinensia, 1710 (Duo to
G. W. Leibniz) 198
Conclusions 199
B. Topical Survey of the Use of Notations 200-356
Signs of Addition and Subtraction 200-216
Early Symbols 200
Origin and Meaning of the Signs 201-3
Spread of the + and — Symbols 204
Shapes of the + Sign 205-7
Varieties of - Signs 208, 209
Symbols for " Plus or Minus" 210,211
Certain Other Specialized Uses of + and — . . 212-14
Four Unusual Signs 215
Composition of Ratios 216
Signs of Multiplication 217-34
Early Symbols 217
Early Uses of the St. Andrew's Cross, but Not as the
Symbol of Multiplication of Two Numbers . . 218-30
The Process of Two False Positions .... 219
Compound Proportions with Integers .... 220
Proportions Involving Fractions 221
Addition and Subtraction of Fractions . . . 222
Division of Fractions 223
Casting Out the 9's, 7's, or ll's 225
Multiplication of Integers 226
Reducing Radicals to Radicals of the Same Order 227
Marking the Place for " Thousands" .... 228
Place of Multiplication Table above 5X5 . . 229
The St. Andrew's Cross Used as a Symbol of Multi-
plication 231
Unsuccessful Symbols for Multiplication . . . 232
The Dot for Multiplication 233
The St. Andrew's Cross in Notation for Transfinite
Ordinal Numbers 234
Signs of Division and Ratio 235-47
Early Symbols 235,236
Rahn's Notation 237
Leibniz's Notations 238
Relative Position of Divisor and Dividend ... 241
Order of Operations in Terms Containing Both -f-
and X 242
A Critical Estimate of : and •§- as Symbols . . 243
TABLE OF CONTENTS
PABAQRAPH8
Notations for Geometric Ratio 244
Division in the Algebra of Complex Numbers . . 247
Signs of Proportion 248-50
Arithmetical and Geometrical Progression . . . 248
Arithmetical Proportion 249
Geometrical Proportion 250
OughtrecTs Notation 251
Struggle in England between Oughtred's and Wing's
Notations before 1700 252
Struggle in England between Oughtred's and Wing's
Notations during 1700-1750 253
Sporadic Notations 254
Oughtred's Notation on the European Continent . 255
Slight Modifications of Oughtred's Notation . . 257
The Notation : : : : in Europe and America . . 258
The Notation of Leibniz 259
Signs of Equality 260-70
Early Symbols 260
Recorde's Sign of Equality 261
Different Meanings of = 262
Competing Symbols 263
Descartes' Sign of Equality 264
Variations in the Form of Descartes' Symbol . . 265
Struggle for Supremacy 266
Variation in the Form of Recorde's Symbol . . . 268
Variation in the Manner of Using It 269
Nearly Equal 270
Signs of Common Fractions 271-75
Early Forms 271
The Fractional Line 272
Special Symbols for Simple Fractions 274
TheSolidus 275
Signs of Decimal Fractions 276-89
Stevin's Notation 276
Other Notations Used before 1617 278
Did Pitiscus Use the Decimal Point? .... 279
Decimal Comma and Point of Napier .... 282
Seventeenth-Century Notations Used after 1617 . 283
Eighteenth-Century Discard of Clumsy Notations . 285
Nineteenth Century : Different Positions for Point
and for Comma 286
Signs for Repeating Decimals 289
Signs of Powers 290-315
General Remarks 290
TABLE OP CONTENTS »
PARAGRAPHS
Double Significance of R and I 291
Facsimiles of Symbols in Manuscripts .... 293
Two General Plans for Marking Powers .... 294
Early Symbolisms: Abbreviative Plan, Index Plan 295
Notations Applied Only to an Unknown Quantity,
the Base Being Omitted 296
Notations Applied to Any Quantity, the Base Being
Designated 297
Descartes' Notation of 1637 298
Did Stampioen Arrive at Descartes' Notation Inde-
pendently? 299
Notations Used by Descartes before 1637 . . . 300
Use of H6rigone's Notation after 1637 .... 301
Later Use of Hume's Notation of 1636 .... 302
Other Exponential Notations Suggested after 1637 . 303
Spread of Descartes' Notation 307
Negative, Fractional, and Literal Exponents . . 308
Imaginary Exponents 309
Notation for Principal Values 312
Complicated Exponents 313
D. F. Gregory's (+)r 314
Conclusions , 315
Signs for Roots 316-38
Early Forms, General Statement 316, 317
The Sign $, First Appearance 318
Sixteenth-Century Use of /J 319
Seventeenth-Century Use of # 321
The Sign I 322
Napier's Line Symbolism 323
The Sign V 324-38
Origin of V 324
Spread of the V 327
Rudolff's Signs outside of Germany .... 328
Stevin's Numeral Root-Indices ...... 329
Rudolff and Stifel's Aggregation Signs . . . 332
Descartes' Union of Radical Sign and Vinculum . 333
Other Signs of Aggregation of Terms . . .. . 334
Redundancy in the Use of Aggregation Signs . 335
Peculiar Dutch Symbolism 336
Principal Root- Values 337
Recommendation of the U.S. National Committee 338
Signs for Unknown Numbers 339-41
Early Forms 339
xii TABLE OF CONTENTS
PARAGRAPHS
Crossed Numerals Representing Powers of Un-
knowns . 340
Descartes' 2, y, x 340
Spread of Descartes' Signs 341
Signs of Aggregation 342-56
Introduction 342
Aggregation Expressed by Letters 343
Aggregation Expressed by Horizontal Bars or Vincu-
lums 344
Aggregation Expressed by Dots 348
Aggregation Expressed by Commas 349
Aggregation Expressed by Parentheses .... 350
Early Occurrence of Parentheses 351
Terms in an Aggregate Placed in a Verbal Column 353
Marking Binomial Coefficients 354
Special Uses of Parentheses 355
A Star to Mark the Absence of Terms .... 356
IV. SYMBOLS IN GEOMETRY (ELEMENTARY PART) 357-85
A, Ordinary Elementary Geometry 357
Early Use of Pictographs 357
Signs for Angles 360
Signs f or " Perpendicular" 364
Signs for Triangle, Square, Rectangle, Paiiillclogram . 365
The Square as an Operator 366
Sign for Circle 367
Signs for Parallel Lines 368
Signs for Equal and Parallel 369
Signs for Arcs of Circles 370
Other Pictographs 371
Signs for Similarity and Congruence 372
The Sign O for Equivalence 375
Lettering of Geometric Figures 376
Sign for Spherical Excess 380
Symbols in the Statement of Theorems 381
Signs for Incommensurables 382
Unusual Ideographs in Elementary Geometry . . . 383
Algebraic Symbols in Elementary Geometry . . . 384
B. Past Struggles between Symbolists and Rhetoricians in
Elementary Geometry .385
INDEX
ILLUSTRATIONS
FIQURB PARAGRAPHS
1. BABYLONIAN TABLETS OF NIPPUR 4
2. PRINCIPLE OF SUBTRACTION IN BABYLONIAN NUMERALS ... 9
3. BABYLONIAN LUNAR TABLES 11
4. MATHEMATICAL CUNEIFORM TABLET CBS 8536 IN THE MUSEUM
OF THE UNIVERSITY OF PENNSYLVANIA 11
5. EGYPTIAN NUMERALS 17
6. EGYPTIAN SYMBOLISM FOR SIMPLE FRACTIONS 18
7. ALGEBRAIC EQUATION IN AHMES 23
8. HIEROGLYPHIC, HIERATIC, AND COPTIC NUMERALS 24
9. PALMYRA (SYRIA) NUMERALS 27
10. SYRIAN NUMERALS 28
11. HEBREW NUMERALS 30
12. COMPUTING TABLE OF SALAMIS 36
13. ACCOUNT OF DISBURSEMENTS OF THE ATHENIAN STATE, 418-
415 B.C. 36
14. ARABIC ALPHABETIC NUMERALS 45
15. DEGENERATE FORMS OF ROMAN NUMERALS 56
16. QUIPU FROM ANCIENT CHANCAY IN PERU 65
17. DIAGRAM OF THE Two RIGHT-HAND GROUPS 65
18. AZTEC NUMERALS 66
19. DRESDEN CODEX OF MAYA 67
20. EARLY CHINESE KNOTS IN STRINGS, REPRESENTING NUMERALS . 70
21. CHINESE AND JAPANESE NUMERALS 74
22. HILL'S TABLE OF BOETHIAN APICES 80
23. TABLE OF IMPORTANT NUMERAL FORMS 80
24. OLD ARABIC AND HINDU-ARABIC NUMERALS 83
25. NUMERALS OF THE MONK NEOPHYTOS 88
26. CHR. RUDOLFF'S NUMERALS AND FRACTIONS 89
27. A CONTRACT, MEXICO CITY, 1649 93
xiv ILLUSTRATIONS
FIGURE PARAGRAPHS
28. REAL ESTATE SALE, MEXICO CITY, 1718 . 94
29. FANCIFUL HYPOTHESES 96
30. NUMERALS DESCRIBED BY NOVIOMAGUS 98
31. SANSKRIT SYMBOLS FOR THE UNKNOWN 108
32. BAKHSHALI ARITHMETIC 109
33. SRIDHARA'S Trisdtika 112
34. ORESME'S Algorismus Proportionum 123
35. AL-QALASADI'S ALGEBRAIC SYMBOLS 125
36. COMPUTATIONS OF REGIOMONTANUS 127
37. CALENDAR OF REGIOMONTANUS 128
38. FROM EARLIEST PRINTED ARITHMETIC 128
39. MULTIPLICATIONS IN THE" TREVISO" ARITHMETIC 128
40. DE LA ROCHE'S Larismethique, FOLIO 605 132
41. DE LA ROCHE'S Larismethique, FOLIO 66A 132
42. PART OF PAGE IN PACIOLI'S Summa, 1523 138
43. MARGIN OF FOLIO 1235 IN PACIOLI'S Summa 139
44. PART OF FOLIO 72 OF GHALIGAI'S Practica d'arithmetica, 1552 . 139
45. GHALIGAI'S Practica d'arithmetica, FOLIO 198 139
46. CARDAN, Ars magna, ED. 1663, PAGE 255 141
47. CARDAN, Ars magna, ED. 1663, PAGE 297 141
48. FROM TARTAGLIA'S General Trattato, 1560 143
49. FROM TARTAGLIA'S General Trattato, FOLIO 4 144
50. FROM BOMBELLI'S Algebra, 1572 144
51. BOMBELLI'S Algebra (1579 IMPRESSION), PAGE 161 .... 145
52. FROM THE MS OF BOMBELLI'S Algebra IN THE LIBRARY OF BOLOGNA 145
53. FROM PAMPHLET No. 595AT IN THE LIBRARY OF THE UNIVERSITY
OF BOLOGNA 146
54. WIDMAN'S Rechnung, 1526 146
55. FROM THE ARITHMETIC OF GRAMMATEUS 146
56. FROM THE ARITHMETIC OF GRAMMATEUS, 1535 147
57. FROM THE ARITHMETIC OF GRAMMATEUS, 1518(?) 147
58. FROM CHR. RUDOLFF'S Coss, 1525 148
ILLUSTRATIONS xv
PARAGRAPHS
59. FROM CHR. RUDOLFF'S Coss, Ev 148
' 60. FROM VAN DER HOECKE' In arithmetica 150
61. PART OF PAGE FROM STIFEL'S Arithmetica intcgra, 1544 . . . 150
62. FROM STIFEL'S Arithmetica Integra, FOLIO 31B 152
63. FROM STIFEL'S EDITION OF RUDOLFF'S Coss, 1553 156
64. SCHEUBEL, INTRODUCTION TO EUCLID, PAGE 28 159
65. W. KLEBITIUS, BOOKLET, 1565 161
66. FROM GLAVIUS' Algebra, 1608 161
67. FROM S. STEVIN'S Le Thiende, 1585 162
68. FROM S. STEVIN'S Arithmetiqve 162
69. FROM S. STEVIN'S Arithmetiqve 164
70. FROM AUREL'S Arithmetica 165
71. R. RECORDS, Whetstone of Witte, 1557 168
72. FRACTIONS IN RECORDE 168
73. RADICALS IN RECORDE 168
74. RADICALS IN DEE'S PREFACE 169
75. PROPORTION IN DEE'S PREFACE 169
76. FROM DIGGES'S Stratioticos 170
77. EQUATIONS IN DIGGES 172
78. EQUALITY IN DIGGES 172
79. FROM THOMAS MASTERSON'S Arithrneticke, 1592 172
80. J. PELETIER'S Algebra, 1554 172
81. ALGEBRAIC OPERATIONS IN PELETIER'S Algebra 172
82. FROM J. BUTEON, Arithmetica, 1559 173
83. GOSSELIN'S De arte magna, 1577 174
84. VIETA, In artem analyticam, 1591 176
85. VIETA, De emendatione aeqvationvm 178
86. B. CAVALIERI, Exercitationes, 1647 179
87. FROM THOMAS HARRIOT, 1631, PAGE 101 189
88. FROM THOMAS HARRIOT, 1631, PAGE 65 189
89. FROM HERIGONE, Cursus mathematicus, 1644 189
90. ROMAN NUMERALS FOR x IN J. HUME, 1635 191
xvi ILLUSTRATIONS
FIGURE PARAGRAPHS
91. RADICALS IN J. HUME, 1635 191
92. R. DESCARTES, Gtomttrie 191
93. I. BARROW'S Euclid, LATIN EDITION. NOTES BY ISAAC NEWTON . 193
94. I. BARROW'S Ewlid, ENGLISH EDITION 193
95. RICH. RAWLINSON'S SYMBOLS 194
96. RAHN'S Teutsche Algebra, 1659 195
97. BRANCKER'S TRANSLATION OF RAHN, 1668 195
98. J. WALLIS, 1657 195
99. FROM THE HIEROGLYPHIC TRANSLATION OF THE AHMES PAPYRUS 200
100. MINUS SIGN IN THE GERMAN MS C. 80, DRESDEN LIBRARY . . 201
101. PLUS AND MINUS SIGNS IN THE LATIN MS C. 80, DRESDEN
LIBRARY 201
102. WIDMANS' MARGINAL NOTE TO MS C. 80, DRESDEN LIBRARY . 201
103. FROM THE ARITHMETIC OF BOETHIUS, 1488 250
104. SIGNS IN GERMAN MSS AND EARLY GERMAN BOOKS .... 294
105. WRITTEN ALGEBRAIC SYMBOLS FOR POWERS FROM PEREZ DE
MOYA'S Arithmetica 294
106. E. WARING'S REPEATED EXPONENTS 313
INTRODUCTION
In this history it has been an aim to give not only the first appear-
ance of a symbol and its origin (whenever possible), but also to indi-
cate the competition encountered and the spread of the symbol among
writers in different countries. It is the latter part of our program
which has given bulk to this history.
The rise of certain symbols, their day of popularity, and their
eventual decline constitute in many cases an interesting story. Our
endeavor has been to do justice to obsolete and obsolescent notations,
as well as to those which have survived and enjoy the favor of mathe-
maticians of the present moment.
If the object of this history of notations were simply to present an
array of facts, more or less interesting to some students of mathe-
matics— if, in other words, this undertaking had no ulterior motive —
then indeed the wisdom of preparing and publishing so large a book
might be questioned. But the author believes that this history consti-
tutes a mirror of past and present conditions in mathematics which
can be made to bear on the notational problems now confronting
mathematics. The successes and failures of the past will contribute to
a more speedy solution of the notational problems of the present time. |
n
NUMERAL SYMBOLS AND COMBINATIONS OF
SYMBOLS
BABYLONIANS
1. In the Babylonian notation of numbers a vertical wedge Y
stood for 1, while the characters ^ and Y>- signified 10 and 100,
respectively. Grotefend1 believes the character for 10 originally to
have been the picture of two hands, as held in prayer, the palms being
pressed together, the fingers close to each other, but the thumbs thrust
out. Ordinarily, two principles were employed in the Babylonial no-
tation— the additive and multiplicative. We shall see that limited use
was made of a third principle, that of subtraction.
2. Numbers below 200 were expressed ordinarily by symbols
whose respective values were to be added. Thus, Y^XKYYY stands
for 123. The principle of multiplication reveals itself in < |>- where
the smaller symbol 10, placed before the 100, is to be multiplied by
100, so that this symbolism designates 1,000.
3. These cuneiform symbols were probably invented by the early
Sumerians. Their inscriptions disclose the use of a decimal scale of
numbers and also of a sexagesimal scale.2
Early Sumerian clay tablets contain also numerals expressed by
circles and curved signs, made with the blunt circular end of a stylus,
the ordinary wedge-shaped characters being made with the pointed
end. A circle • stood for 10, a semicircular or lunar sign stood for 1.
Thus, a "round-up" of cattle shows J*DDD> or ^ cows.3
4. The sexagesimal scale was first discovered on a tablet by E.
Hincks4 in 1854. It records the magnitude of the illuminated portion
1 His first papers appeared in Gottingische Gelehrte Anzeigen (1802), Stuck 149
und 178; ibid. (1803), Stuck 60 und 117.
2 In the division of the year and of the day, the Babylonians used also the
duodecimal plan.
8 G. A. Barton, Haverford Library Collection of Tablets, Part I (Philadelphia,
1905), Plate 3, HCL 17, obverse; see also Plates 20, 26, 34, 35. Allotte de la
Fuye, "En-e-tar-zi pate*si de Lagas," H. V. Hilprecht Anniversary Volume (Chi-
cago, 1909), p. 128, 133.
4 "On the Assyrian Mythology," Transactions of the Royal Irish Academy.
"Polite Literature," Vol. XXII, Part 6 (Dublin, 1855), p. 406, 407.
2
OLD NUMERAL SYMBOLS 3
of the moon's disk for every day from new to full moon, the whole disk
being assumed to consist of 240 parts. The illuminated parts during
the first five days are the series 5, 10, 20, 40, 1.20, which is a geo-
metrical progression, on the assumption that the last number is 80.
From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8,
2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the common difference being 16.
The last number is written in the tablet X^,— and, according to
Hincks's interpretation, stood for 4 X 60 = 240.
Obverse. Reverse.
FIG. 1. — Babylonian tablets from Nippur, about 2400 B.C.
5, Hincks's explanation was confirmed by the decipherment of
tablets found at Senkereh, near Babylon, in 1854, and called the Tab-
lets of Senkereh. One tablet was found to contain a table of square
numbers, from I2 to 602, a second one a table of cube numbers from I3
to 323. The tablets were probably written between 2300 and 1600 B.C.
Various scholars contributed toward their interpretation. Among
them \vere George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr.
Lenormant, and finally R. Lepsius.1 The numbers 1, 4, 9, 16, 25, 36,
George Smith, North British Review (July, 1870), p. 332 n.; J. Oppert,
Journal asiatique (August-September, 1872; October-November, 1874); J.
Oppert, £talon des tnesures assyr. fixe" par les textes cuneiformes (Paris, 1874) ; Sir
H. Rawlinson and G. Smith, "The Cuneiform Inscriptions of Western Asia,"
Vol. IV: A Selection from the Miscellaneous Inscriptions of Assyria (London,
1875), Plate 40; R. Lepsius, "Die Babylonisch-Assyrischen Langenmaasse nach
der Tafel von Senkereh," Abhandlungen der Koniglichen Akademie der Wissen-
schaften zu Berlin (aus dem Jahre 1877 [Berlin, 1878], Philosophisch-historische
Klasse), p. 105-44.
4 A HISTORY OF MATHEMATICAL NOTATIONS
and 49 are given as the squares of the first seven integers, respecti
We have next 1.4 = 82, 1.21 = 92, 1.40= 102, etc. This clearly indi
the use of the sexagesimal scale which makes 1.4 = 60+4, 1.21 =
21. 1.40 = 60+40, etc. This sexagesimal system marks the ea:
appearance of the all-important "principle of position" in wr
numbers. In its general and systematic application, this principl
quires a symbol for zero. But no such symbol has been found on <
Babylonian tablets; records of about 200 B.C. give a symbol for
as we shall see later, but it was not used in calculation. The ea:
thorough and systematic application of a symbol for zero anc
principle of position was made by the Maya of Central America, a
the beginning of the Christian Era.
6. An extension of our knowledge of Babylonian mathem
was made by H. V. Hilprecht who made excavations at Nuffar
ancient Nippur). We reproduce one of his tablets1 in Figure 1.
Hilprecht's transliteration, as given on page 28 of his te
as follows:
Line 1. 125 720 Line 9. 2,000
Line 2. IGI-GAL-BI 103,680 Line 10. IGI-GAL-BI (
Line 3. 250 360 Line 11. 4,000
Line 4. IGI-GAL-BI 51,840 Line 12. IGI-GAL-BI
Line 5. 500 180 Line 13. 8,000
Line 6. IGI-GAL-BI 25,920 Line 14. IGI-GAL-BI
Line 7. 1,000 90 Line 15. 16,000
Line 8. IGI-GAL-BI 12,960 Line 16. IGI-GAL-BI
7. In further explanation, observe that in
Line 1. 125 = 2X60+5, 720 = 12X60+0
Line 2. Its denominator, 103,680 = [28 X60+48(?)]X 6
Line 3. 250 = 4X60+10, 360 = 6X60+0
Line 4. Its denominator, 51,840 = [14 X 60+24] X60+(
Line 5. 500 = 8X60+20, 180 = 3X60+0
Line 6. Its denominator, 25,920 = [7 X 60+ 12] X 60+0
Line 7. 1,000=16X60+40, 90=1X60+30
Line 8. Its denominator, 12,960 = [3X60+36]X60+0
1 The Babylonian Expedition of the University of Pennsylvania. Seri
"Cuneiform Texts," Vol. XX, Part 1, Mathematical, Metrological and C)
logical Tablets from the Temple Library of Nippur (Philadelphia, 1906), Pla
No. 25.
OLD NUMERAL SYMBOLS 5
Line 9. 2,000 = 33X60+20, 18=10+8
Line 10. Its denominator, 6,480 = [IX 60+48] X 60+0
Line 11. 4,000 = [1X60+6]X60+40, 9
Line 12. Its denominator, 3,240 = 54 X 60+0
Line 13. 8,000 = [2X60+13]X60+20, 18
Line 14. Its denominator, 1,620 = 27X60+0
Line 15. 16,000 = [4X60+ 26] X 60+40, 9
Line 16. Its denominator, 810=13X60+30
IGI-GAL = Denominator, £/ = Its, i.e., the number 12,960,000 or 604.
We quote from Hilprecht (op. cit., pp. 28-30):
"We observe (a) that the first numbers of all the odd lines (1, 3, 5,
7, 9, 11, 13, 15) form an increasing, and all the numbers of the even
lines (preceded by IGI-GAL-BI = (its denominator') a descending
geometrical progression; (6) that the first number of every odd line
can be expressed by a fraction which has 12,960,000 as its numerator
and the closing number of the corresponding even line as its denomi-
nator, in other words,
10 12,960,000 . 12,960,000 . Knn_ 12,960,000
1^)"" 103,680 ' 51,840 ' 25,920 '
nn 12,960,000 . 9 mn 12,960,000 . 12,
>m = -l2W- ' 2>°00= 6480 ' 4'000== 3
960,000
6,480 ' ' 3,240 '
12,960,000 . 12,960,000
8,000= lj62Q , 16,000-—^—.
But the closing numbers of all the odd lines (720, 360, 180, 90, 18, 9,
18, 9) are still obscure to me .....
"The question arises, what is the meaning of all this? What in par-
ticular is the meaning of the number 12,960,000 ( = 604 or 3,6002)
which underlies all the mathematical texts here treated ....?....
This ' geometrical number ' (12,960,000), which he [Plato in his Repub-
lic viii. 546#-D] calls 'the lord of better and worse births/ is the
arithmetical expression of a great law controlling the Universe.
According to Adam this law is 'the Law of Change, that law of in-
evitable degeneration to which the Universe and all its parts are sub-
ject' — an interpretation from which I arn obliged to differ. On the
contrary, it is the Law of Uniformity or Harmony, i.e. that funda-
mental law which governs the Universe and all its parts, and which
cannot be ignored and violated without causing an anomaly, i.e. with-
out resulting in a degeneration of the race." The nature of the "Pla-
tonic number" is still a debated question.
6 A HISTORY OF MATHEMATICAL NOTATIONS
8. In the reading of numbers expressed in the Babylonian sexa-
gesimal system, uncertainty arises from the fact that the early Baby-
lonians had no symbol for zero. In the foregoing tablets, how do we
know, for example, that the last number in the first line is 720 and
not 12? Nothing in the symbolism indicates that the 12 is in the place
where the local value is "sixties" and not "units." Only from the
study of the entire tablet has it been inferred that the number in-
tended is 12X60 rather than 12 itself. Sometimes a horizontal line
was drawn following a number, apparently to indicate the absence
of units of lower denomination. But this procedure was not regular,
nor carried on in a manner that indicates the number of vacant places.
9. To avoid confusion some Babylonian documents even in early
times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600,
36,000.* Thus • was 10, • was 3,600, © was 36,000.
in view of other variants occurring in fchc
mathematical tablets from Nippur, notably the numerous variants of "19,"1 some of
which may be merely scribal errors :
They evidently all go back to the form <^}~ ^^^f (20 — 1 = 19).
FIG. 2. — Showing application of the principle of subtraction
10. Besides the principles of addition and multiplication, Baby-
lonian tablets reveal also the use of the principle of subtraction, which
is familiar to us in the Roman notation XIX (20—1) for the number
19. Hilprecht has collected ideograms from the Babylonian tablets
which he has studied, which represent the number 19. We reproduce
his symbols in Figure 2. In each of these twelve ideograms (Fig. 2),
the' two symbols to the left signify together 20. Of the symbols im-
mediately to the right of the 20, one vertical wedge stands for "one"
and the remaining symbols, for instance Y^, for LAL or "minus";
the entire ideogram represents in each of the twelve cases the number
20- lor 19.
One finds the principle of subtraction used also with curved
signs;2 D • • Y*~~D meant 60+20 — 1, or 79.
1 See Frangois Thureau-Dangin, Recherches sur Vorigine de Vecriture cuntiforme
(Paris, 1898), Nos. 485-91, 509-13. See also G. A. Barton, Haverford College
Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905), where the
forms are somewhat different; also the Hilprecht Anniversary Volume (Chicago,
1909), p. 128 ff.
2 G. A. Barton, op. cit.t Plate 3, obverse.
OLD NUMERAL SYMBOLS
11. The symbol used about the second century B.C. to designate
5 absence of a number, or a blank space, is shown in Figure 3, con-
ning numerical data relating to the moon.1 As previously stated,
s symbol, ^ , was not used in computation and therefore performed
FIG. 3. — Babylonian lunar tables, reverse; full moon for one year, about the
1 of the second century B.C.
ly a small part of the functions of our modern zero. The symbol is
jn in the tablet in row 10, column 12; also in row 8, column 13.
igler's translation of the tablet, given in his book, page 42, is shown
low. Of the last column only an indistinct fragment is preserved;
3 rest is broken off.
REVERSE
Niaannu
28°56'30"
19°16' " Librae
3Z 6°45'
4i74ii10ur sik
Airu
28 38 30
175430 Scorpii
321 28
620 30 sik
Simannu
28 20 30
16 15 Arcitenentia
3 31 39
345 30 sik
Dti,zu
28 18 30
14 33 30 Capri
33441
1 10 30 sik
Abu
28 36 30
13 9 Aquarii
32756
1 24 30 bar
Ululu
29 54 30
13 3 30 Piscium
3 1534
1 59 30 num
TiSrltu
29 12 30
11 16 Arietis
258 3
4 34 30 num
Araty-s.
29 30 30
10 46 30 Tauri
24054
6 0 10 num
Kishmu
29 48 30
10 35 Geminorurn
2 29 29
3 25 10 num
Tebitu
29 57 30
10 32 30 Cancri
22430
067 10 num
Sabatu
29 39 30
10 12 Leonis
2 30 53
1 44 50 bar
Addru I
29 21 30
9 33 30 Virginis
2 42 56
2 19 50 sik
Ad6.ru II
29 330
8 36 Librae
3 021
464 50 sik
Nisannu
28 45 30
7 21 30 Scorpii
3 1736
539 50 sik
1 Franz Xaver Kugler, S. J., Die babylonische Mondrechnung (Freiburg im Breis-
j, 1900), Plate IV, No. 99 (81-7-6), lower part.
A HISTORY OF MATHEMATICAL NOTATIONS
OBVERSE
W^r-
4
«
^r
EMte-
_^MN
w^
W*-
s^-
31
^¥^-
(THff^-
El^Si^
^^
r-«w-'n..*'
^teh'^tfilii
FIG. 4. —Mathematical cuneiform tablet, CBS 8536, in the Museum of the
University of Pennsylvania.
OLD NUMERAL SYMBOLS 9
12. J. Oppert pointed out the Babylonian use of a designation
the sixths, viz., |, £, -£, f, $. These are unit fractions or fractions
ose numerators are one less than the denominators.1 He also ad-
iced evidence pointing to the Babylonian use of sexagesimal frac-
is and the use of the sexagesimal system in weights and measures.
B occurrence of sexagesimal fractions is shown in tablets recently
mined. We reproduce in Figure 4 two out of twelve columns found
a tablet described by H. F. Lutz.2 According to Lutz, the tablet
innot be placed later than the Cassite period, but it seems more prob-
e that it goes back even to the First Dynasty period, ca. 2000 B.C."
13, To mathematicians the tablet is of interest because it reveals
orations with sexagesimal fractions resembling modern operations
h decimal fractions. For example, 60 is divided by 81 and the
>tient expressed sexagesimally. Again, a sexagesimal number with
> fractional places, 44 (26) (40), is multiplied by itself, yielding a
.duct in four fractional places, namely, [32]55(18)(31)(6)(40). In
3 notation the [32] stands for 32X60 units, and to the (18), (31),
, (40) must be assigned, respectively, the denominators 60, 602,
, 604.
The tablet contains twelve columns of figures. The first column
g. 4) gives the results of dividing 60 in succession by twenty-nine
'erent divisors from 2 to 81. The eleven other columns contain
les of multiplication; each of the numbers 50, 48, 45, 44 (26) (40),
36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by
numbers 30, 40, 50, and finally by itself. Using our modern nu-
rals, we interpret on page 10 the first and the fifth columns. They
dbit a larger number of fractions than do the other columns.
e Babylonians had no mark separating the fractional from the in-
ral parts of a number. Hence a number like 44 (26) (40) might be
3rpreted in different ways; among the possible meanings are 44 X
+26X60+40, 44X60+26+40X60-1, and 44+26X60~x+40X
-2. Which interpretation is the correct one can be judged only by
context, if at all.
The exact meaning of the first two lines in the first column is un-
fcain. In this column 60 is divided by each of the integers written
the left. The respective quotients are placed on the right.
1 Symbols for such fractions are reproduced also by Thureau-Dangin, op. cit.,
\. 481-84, 492-508, and by G. A. Barton, Haverford College Library Collection
Cuneiform Tablets, Part I (Philadelphia, 1905).
2 "A Mathematical Cuneiform Tablet/' American Journal of Semitic Lan-
ges and Literatures, Vol. XXXVI (1920), p. 24^-57.
10
A HISTORY OF MATHEMATICAL NOTATIONS
In the fifth column the multiplicand is 44 (26) (40) or 44 jj.
The last two lines seem to mean "602-r-44(26)(40) = 81, 602-r81 =
44(26)(40)."
First Column
.... gal (?) -bi 40 -&m
Su a- na gal-bi 30 -am
igi 2
30
igi 3
20
igi 4
15
igi 5
12
igi 6
10
igi 8
7(30)
igi 9
6(40)
igi 10
6
igi 12
5
igi 15
4
igi 16
•3(45)
igi 18
3(20)
igi 20
3
igi 24
2(30)
igi 25
2(24)
igi 28*
2(13) (20)
igi 30
2
igi 35*
1(52) (30)
igi 36
1(40)
igi 40
1(30)
igi 45
1(20)
igi 48
1(15)
igi 50
1(12)
igi 54
1(6) (40)
igi 60
1
igi 64
(56) (15)
igi 72
(50)
igi 80
(45)
igi 81
(44) (26) (40)
1
2
3
4
5
6
7
9
10
11
12
13
14
15
16
17
18
19
20
30
40
50
Fifth Column
44(26) (40)
44(26) (40)
[1]28(53)(20)
[2]13(20)
[2]48(56)(40)*
[3]42(13)(20)
[4]26(40)
[6]40
[7]24(26)(40)
[8]8(53)(20)
[8]53(20)
[9]27(46)(40)*
[10]22(13)(20)
[H]6(40)
[12]35(33)(20)
[13J20
[14]4(26)(40)
[14]48(53)(20)
[22]13(20)
[29]37(46)(40)
[38]2(13)(20)*
44(26) (40)a-na 44(26) (40)
[32]55(18)(31)(6)(40)
44 (26) (40) square
igi 44(26)(40) 81
igiSl 44 (26) (40)
Numbers that are incorrect are marked by an asterisk (*).
14. The Babylonian use of sexagesimal fractions is shown also in
a clay tablet described by A. Ungnad.1 In it the diagonal of a rec-
tangle whose sides are 40 and 10 is computed by the approximation
1 Orientalische Literaturzeitung (ed. Peise, 1916), Vol. XIX, p. 363-68. See
also Bruno Meissner, Babylonien und Assyrien (Heidelberg, 1925), Vol. II, p. 393.
OLD NUMERAL SYMBOLS 11
40+2X40Xl02-h602, yielding 42(13)(20), and also by the approxi-
mation 40+1024- 12X401, yielding 41(15). Translated into the deci-
mal scale, the first answer is 42.22+, the second is 41.25, the true
value being 41.23+. These computations are difficult to explain,
except on the assumption that they involve sexagesimal fractions.
15. From what has been said it appears that the Babylonians had
ideograms which, transliterated, are Igi-Gal for "denominator" or
"division," and Lai for "minus." They had also ideograms which,
transliterated, are Igi-Dua for "division," and A-Du and Ara for
"times," as in Ara- 1 18, for "1X18 = 18," Ara- 2 36 for
"2 X 18 = 36" ; the Ara was used also in "squaring," as in 3 Ara 3 9
for "3X3 = 9." They had the ideogram Ba-Di-E for "cubing," as
in 21 -E 3 Ba-Di-E for "33 = 27"; also Ib-Di for "square," as in 9-#
3 Ib-Di for "32 = 9." The sign A -An rendered numbers "distribu-
tive."1
EGYPTIANS
16. The Egyptian number system is based on the scale of 10, al-
though traces of other systems, based on the scales of 5, 12, 20, and
60, are believed to have been discovered.2 There are three forms of
Egyptian numerals: the hieroglyphic, hieratic, and demotic. Of these
the hieroglyphic has been traced back to about 3300 B.C. ;3 it is found
mainly on monuments of stone, wood, or metal. Out of the hiero-
glyphic sprang a more cursive writing known to us as hieratic. In the
beginning the hieratic was simply the hieroglyphic in the rounded
forms resulting from the rapid manipulation of a reed-pen as con-
trasted with the angular and precise shapes arising from the use of the
chisel. About the eighth century B.C. the demotic evolved as a more
abbreviated form of cursive writing. It was used since that time down
to the beginning of the Christian Era. The important mathematical
documents of ancient Egypt were written on papyrus and made use of
the hieratic numerals.4
1 Hilprecht. op. til., p. 23; Arno Poebel, Grundzuge der sumerischen Grammatik
(Rostock, 1923), p. 115; B. Meissner, op. cit., p. 387-89.
2 Kurt Sethe, Von ZdhLen und Zahlworlen bei den alien Agyptern (Strassburg,
1916), p. 24-29.
3 J. E. Quibell and F. W. Green, Hierakonopolis (London, 1900-1902), Part I,
Plate 26B, who describe the victory monument of King Ncr-mr; the number of
prisoners taken is given as 120,000, while 400,000 head of cattle and 1,422,000
goats were captured.
4 The evolution of the hieratic writing from the hieroglyphic is explained in
G. Moller, Hieratische Palaographie, Vol. I, Nos. 614 ff. The demotic writing
12
A HISTORY OF MATHEMATICAL NOTATIONS
17. The hieroglyphic symbols were I for 1, O for 10, C for 100,
I for 1,000, | for 10,000, ^ for 100,000, $ for 1,000,000, Q for
10,000,000. The symbol for 1 represents a vertical staff; that for
1,000 a lotus plant; that for 10,000 a pointing finger; that for 100,000
a burbot; that for 1,000,000 a man in astonishment, or, as more recent
Etner
Zehner
HunJerte
TctusettJe
n
I
I
60
M
nn
A
II
100
01
no
n
ODD
nnnn
JiL
Ann
KS
not
8100
AAAA
a
i
i
rwvi
nnn
II?
CO
FIG. 5. — Egyptian numerals. Hieroglyphic, hieratic, and demotic numeral
symbols. (This table was compiled by Kurt Sethc.)
Egyptologists claim, the picture of the cosmic deity Hh.1 The sym-
bols for 1 and 10 are sometimes found in a horizontal position.
18. We reproduce in Figures 5 and 6 two tables prepared by Kurt
is explained by F. L. Griffith, Catalogue of the Demotic Papyri in the John Rylands
Library (Manchester, 1909), Vol. Ill, p. 415 if., and by H. Brugsch, Grammaire
d&motique, §§ 131 ff.
1Sethe, op.cit., p. 11, 12.
OLD NUMERAL SYMBOLS
13
Sethe. They show the most common of the great variety of forms which
are found in the expositions given by Moller, Griffith, and Brugsch.
Observe that the old hieratic symbol for % was the cross X, sig-
nifying perhaps a part obtainable from two sections of a body through
the center.
Attaeyyfttiscke BruchiticLcn,
ftrc&ttckt fruchzetJu*
tit
in
M>
IP
X
Mil
mi
III!
**
lift
%'/*
tv
%'At
y<
[llllllj
nun
mm
//*
*;»
-90*
FIG. 6. — Egyptian symbolism for simple fractions. (Compiled by Kurt Sethe)
19. In writing numbers, the Egyptians used the principles of addi-
tion and multiplication. In applying the additive principle, not more
than four symbols of the same kind were placed in any one group.
Thus, 4 was written in hieroglyphs 1 1 1 1 ; 5 was not written HIM, but
either 1 1 1 1 1 or , , . There is here recognized the same need which
caused the Romans to write V after IIII, L = 50 after XXXX = 40,
D = 500 after CCCC = 400. In case of two unequal groups, the Egyp-
tians always wrote the larger group before, or above the smaller group;
thus, seven was written ,, , .
14 A HISTORY OF MATHEMATICAL NOTATIONS
20. In the older hieroglyphs 2,000 or 3,000 was represented by two
or three lotus plants grown in one bush. For example, 2,000 was ^ ;
correspondingly, 7,000 was designated by 23K £?£ . The later hiero-
glyphs simply place two lotus plants together, to represent 2,000, with-
out the appearance of springing from one and the same bush.
21. The multiplicative principle is not so old as the additive; it
came into use about 1600-2000 B.C. In the oldest example hitherto
known,1 the symbols for 120, placed before a lotus plant, signify
120,000. A smaller number written before or below or above a sym-
bol representing a larger unit designated multiplication of the larger by
the smaller. Mollcr cites a case where 2,800,000 is represented by one
burbot, with characters placed beneath it which stand for 28.
22. In hieroglyphic writing, unit fractions were indicated by
placing the symbol <o over the number representing the denomina-
tor. Exceptions to this arc the modes of writing the fractions | and f ;
the old hieroglyph for \ was ^=T, the later was /" ~; of the slightly
varying hieroglyphic forms for -|, £ was quite common.2
23. We reproduce an algebraic example in hieratic symbols, as it
occurs in the most important mathematical document of antiquity
known at the present time — the Rhind papyrus. The scribe, Ahmcs,
who copied this papyrus from an older document, used black and red
ink, the red in the titles of the individual problems and in writing
auxiliary numbers appearing in the computations. The example
which, in the Eisenlohr edition of this papyrus, is numbered 34, is
hereby shown.3 Hieratic writing was from right to left. To facilitate
the study of the problem, we write our translation from right to left
and in the same relative positions of its parts as in the papyrus, except
that numbers are written in the order familiar to us; i.e., 37 is written
in our translation 37, and not 73 as in the papyrus. Ahmes writes
unit fractions by placing a dot over the denominator, except in case of
1 Ibid., p. 8.
2 Ibid., p. 92-97, gives detailed information on the forms representing f.
The Egyptian procedure for decomposing a quotient into unit fractions is explained
by V. V. Bobynin in Abh. Gesch. Math., Vol. IX (1899), p. 3.
8 Ein matkematisches Handbook der alien Agypter (Papyrus Rhind des British
Museum) t'ibersetzt und erkldrt (Leipzig, 1877; 2d cd., 1891). The explanation of
Problem 34 is given on p. 55, the translation on p. 213, the facsimile reproduction
on Plate XIII of the first edition. The second edition was brought out without the
plates. A more recent edition of the Ahmes papyrus is due to T. Eric Peet and
appears under the title The Rhind Mathematical Papyrus, British Museum,
Nos. 10057 and 10058, Introduction, Transcription, and Commentary (London,
1923).
OLD NUMERAL SYMBOLS 15
i> i> £> i> eacn °f which had its own symbol. Some of the numeral
symbols in Ahmes deviate somewhat from the forms given in the two
preceding tables; other symbols are not given in those tables. For the
reading of the example in question we give here the following symbols :
Four — One-fourth X
Five "1 Heap S$t See Fig. 7
Seven Q- The whole |J See Fig. 7
One-half ~7 It gives & See Fig. 7
FIG. 7. — An algebraic equation and its solution in the Ahmes papyrus, 1700
B.C., or, according to recent authorities, 1550 B.C. (Problem 34, Plate XIII in
Eisenlohr; p. 70 in Peet; in chancellor Chace's forthcoming edition, p. 76, as R. C.
Archibald informs the writer.)
Translation (reading from right to left) :
"10 gives it, whole its, \ its, \ its, Heap No. 34
al4 j I 5 is heap the together 7 4
1 I
Proof the of Beginning
-iV 1 1 5
I \- Remainder £ -£ 9 together A izV i i 1 I
14 gives i A -^V •&• TT A I
21 Together .7 gives i 122448"
16
A HISTORY OF MATHEMATICAL NOTATIONS
24. Explanation:
oc or
The algebraic equation is 0+4+2= 10
i.e., (l+i+i)*=10
The solution answers the question, By what must (1 ^ |) be
multiplied to yield the product 10? The four lines 2-5 contain on the
right the following computation :
Twice (1 H) yields 3 fc.
Four times (1 -£ \) yields 7.
One-seventh of (1 \ -J-) is \.
t° UNITES.
. S1GNES
LETTRES
MIMEIULES
copies.
VALEUR
ik-s
S10NES.
NOMS
OK A01IBBK
<m
dialecle tliebaiu.
UlKROCLYriUQUES ,
creux ct plciiis.
UI8AAT1Q1IBS ,
uvcc variantcs.
0 i
') I ? ?
£
t
Vt«l.
00 H
H 'M
&
rt
snau.
DOD HI
^ 04
TT
:t
chonwnt.
DDOD i!
UH -u^ 4
^
A
ftoou.
ODD 00 '"
1 1 1
E
r>
lion.
DOD DOD !!!
t Z
r
(>
soon.
0000000 V"
^t xti ^&i
t.
7
sachf.
flDDOOODO m'i
^=^ =*•
F
8
chinoun.
000 ODD ODD "Si!"
^.^
tf
<)
pxix.
(Continued on facing page]
[i.e., taking (1 \ -J-) once, then four times, together with \ of it, yields
only 9; there is lacking 1. The remaining computation is on the
four lines 2-5, on the left. Since \ of (1 | {) yields (\ -^ ^8) or -J,
lor]
(i A) of (1 i 1), yields |.
And the double of this, namely, (^ -£f) of (1 \ |) yields 1.
Adding together 1, 4, | and (fc iV), we obtain Heap = 5^
^ -^ or 5f , the answer.
OLD NUMERAL SYMBOLS
17
Proof. — 5 ^ \ ilf is multiplied by (1 ^ J) and the partial products
are added. In the first line of the proof we have 5 ^ | ^lf, in the second
line half of it, in the third line one-fourth of it. Adding at first only
the integers of the three partial products and the simpler fractions
i> i> !> i> i> the partial sum is 9 -\ \. This is \ I short of 10. In the
fourth line of the proof (1. 9) the scribe writes the remaining fractions
and, reducing them to the common denominator 56, he writes (in
2° DIZA1NES.
SIGiNES
LETTRES
NUMKRALES
copies.
VALEUR
dcs
8IONK8.
NOMS
DE NOMDRB
en
dialcctc thebaih.
II1EROGLYPI1IQURS .
crcux el plein.
IIIERATIQUES ,
nvcc varianles.
X X /6
I
10
ment.
ChinVe connmm
dos dizaincs :
XX
*
30
sjouoL
maab.
mi »« n
^- -*-
H
/JO
hme.
\ *\
1?
5o
taioii.
Jii &.
I
(>o
se.
°x °x
o"
7°
chfe.
>ui4 jiU
TT
80
hmeiie.
^
Ci
9°
ptsldtou.
FIG. 8. — Hieroglyphic, hieratic, and Coptic numerals. (Taken from A. P.
Pihan, Expos6 des signes de numeration [Paris, 1860], p. 26, 27.)
red color) in the last line the numerators 8, 4, 4, 2, 2, 1 of the reduced
fractions. Their sum is 21. But ew =
5o
^-=-7 o , which is the exact
oo 4 8
amount needed to make the total product 10.
A pair of legs symbolizing addition and subtraction, as found in
impaired form in the Ahmes papyrus, are explained in § 200.
25. The Egyptian Coptic numerals are shown in Figure 8. They
are of comparatively recent date. The hieroglyphic and hieratic are
18 A HISTORY OF MATHEMATICAL NOTATIONS
the oldest Egyptian writing; the demotic appeared later. The Cop-
tic writing is derived from the Greek and demotic writing, and was
used by Christians in Egypt after the third century. The Coptic
numeral symbols were adopted by the Mohammedans in Egypt after
their conquest of that country.
26. At the present time two examples of the old Egyptian solu-
tion of problems involving what we now term "quadratic equations"1
are known. For square root the symbol Ir3 has been used in the modern
hieroglyphic transcription, as the interpretation of writing in the two
papyri; for quotient was used the symbol oo .
PHOENICIANS AND SYRIANS
27. The Phoenicians2 represented the numbers 1-9 by the re-
spective number of vertical strokes. Ten was usually designated by
a horizontal bar. The numbers 1 1-19 were expressed by the juxtaposi-
tion of a horizontal stroke and the required number of vertical ones.
Palmyreaische ZaMzeiebn I X 3; 3, JD'p^ ;,55"7 . '7^3 3 '''CO"
Virianten >ei Oruter / -V ; >. 0; , V; >V( >.V ''^0><?'>V'
BtdeuUag 1. 0; 10. 20 100, 110. 1000 JW.
FIG. 9. — Palmyra (Syria) numerals. (From M. Cantor, Kulturleben, etc., Fig. 48)
As Phoenician writing proceeded from right to left, the horizontal
stroke signifying 10 was placed farthest to the right. Twenty was
represented by two parallel strokes, either horizontal or inclined and
sometimes connected by a cross-line as in H, or sometimes by two
strokes, thus A- One hundred was written thus |<| or thus | £>| . Phoe-
nician inscriptions from which these symbols are taken reach back
several centuries before Christ. Symbols found in Palmyra (modern
Tadmor in Syria) in the first 250 years of our era resemble somewhat
the numerals below 100 just described. New in the Palmyra numer-
1 See H. Schack-Schackcnburg, "Der Berliner Papyrus 6619," Zeitschrift fur
dgyptische Sprache und Altertumskunde, Vol. XXXVIII (1900), p. 136, 138, and
Vol. XL (1902), p. 6S-66.
2 Our account is taken from Moritz Cantor, Vorlesungen fiber Geschichte der
Mathematik, Vol. I (3d ed. ; Leipzig, 1907), p. 123, 124; Mathematische Beitrage zum
KuUurleben der Volker (Halle, 1863), p. 255, 256, and Figs. 48 and 49.
OLD NUMERAL SYMBOLS 19
als is 7 for 5. Beginning with 100 the Palmyra numerals contain new
forms. Placing a I to the right of the sign for 10 (see Fig. 9) signifies
multiplication of 10 by 10, giving 100. Two vertical strokes 1 1 mean
10X20, or 200; three of them, 10X30, or 300.
28. Related to the Phoenician are numerals of Syria, found in
manuscripts of the sixth and seventh centuries A.D. Their shapes and
their mode of combination are shown in Figure 10. The Syrians em-
ployed also the twenty-two letters of their alphabet to represent the
numbers 1-9, the tens 10-90, the hundreds 100-400. The following
hundreds were indicated by juxtaposition: 500 = 400+100, 600 =
400+200, ____ , 900=400+400+100, or else by writing respectively
50-90 and placing a dot over the letter to express that its value is to
be taken tenfold. Thousands were indicated by the letters for 1-9,
with a stroke annexed as a subscript. Ten thousands were expressed
I, H - 2, HI - 3, FP- 4,. -*-5. h-* -6
7 HM-8. H^-9 7-io 7-u K7-12
w, HH^ -18, O - 20 70 - M. TI - 100
Syrische Zahlzeiche.n
FIG. 10. — Syrian numerals. (From M. Cantor, Kulturleben, etc., Fig. 49)
by drawing a small dash below the letters for one's and ten's. Millions
were marked by the letters 1-9 with two strokes annexed as sub-
scripts (i.e., 1,000X1,000 = 1,000,000).
HEBREWS
29. The Hebrews used their alphabet of twenty-two letters for
the designation of numbers, on the decimal plan, up to 400. Figure
11 shows three forms of characters: the Samaritan, Hebrew, and
Rabbinic or cursive. The Rabbinic was used by commentators of the
Sacred Writings. In the Hebrew forms, at first, the hundreds from 500
to 800 were represented by juxtaposition of the sign for 400 and a
second number sign. Thus, pn stood for 500, ^n for 600, ISO for 700,
nn for 800.
30. Later the end forms of five letters of the Hebrew alphabet
came to be used to represent the hundreds 500-900. The five letters
representing 20, 40, 50, 80, 90, respectively, had two forms; one of
20
A HISTORY OF MATHEMATICAL NOTATIONS
LETTRES
MOMS
NOMS
8AMAIUTAIRES
HEBBAIQUES.
BABBI1UQUBS.
DBS LKTTRB3.
DE NOMBBE.
*
N
(S
aleph , a
t
ekhdd.
a
2
5
bet, b
2
chewing
1
:
J
ghimcl , gh
3
clielochdh.
1
1
1
dalet, d
A
arbd'ah.
*
n
p
y, A
5
khamichdh.
*
i
)
waw, w
6
chichdh.
1»
T
t
zain, z
7
chib'dh.
*
n
p
khel, B
8
chemondh.
*
D
V
t'et', t'
9
tich'dh.
m
'
5
iod, t
10
'asdrdh.
a
3
o
kaph , k
30
'esrim.
4
h
1)
lamed , /
do
chelochim*
»
D
p
mem , m
%0
arbd'im.
A
J
D
noun y n
5o
khamichim.
*
D
D
s'amek i
60
chichim.
v
y
J>
cain , '«
70
chib'im.
3
s
D
ph^, ph
80
chemonim.
Ytt
3J
3
Lsade, to
9<>
tictiim.
1?
p
P
qopli, ^
100
mtdh.
^
*]
•5
rech, r
900
mdtai'm.
JJJL
t^
ft
chin , ch
3oo
cluilvchmttt.
A
n
P
lau, £
W400
arba* mcdt.
FIG. 11. — Hebrew numerals. (Taken from A. P. Pihan, Expos6 des signes de
numeration [Paris, 1860], p. 172, 173.)
OLD NUMERAL SYMBOLS 21
the forms occurred when the letter was a terminal letter of a word.
These end forms were used as follows:
Y T| T D 1
900 800 700 600 500.
To represent thousands the Hebrews went back to the beginning of
their alphabet and placed two dots over each letter. Thereby its
value was magnified a thousand fold. Accordingly, £ represented
1,000. Thus any number less than a million could be represented by
their system.
31. As indicated above, the Hebrews wrote from right to left.
Hence, in writing numbers, the numeral of highest value appeared on
the right; )$n meant 5,001, n& meant 1,005. But 1,005 could be
written also flK , where the two dots were omitted, for when ^ meant
unity, it was always placed to the left of another numeral. Hence
when appearing on the right it was interpreted as meaning 1,000.
With a similar understanding for other signs, one observes here the
beginning of an imperfect application in Hebrew notation of the
principle of local value. By about the eighth century A.D., one finds
that the signs iTD^n signify 5,845, the number of verses in the laws
as given in the Masora. Here the sign on the extreme right means
5,000; the next to the left is an 8 and must stand for a value less than
5,000, yet greater than the third sign representing 40. Hence the
sign for 8 is taken here as 800. l
GREEKS
32. On the island of Crete, near Greece, there developed, under
Egyptian influence, a remarkable civilization. Hieroglyphic writing
on clay, of perhaps about 1500 B.C., discloses number symbols as
follows: ) or I for 1, ))))) or 1 1 1 1 1 or "' for 5, • for 10, \ or / for
100, <> for 1,000, V for i (probably), \\\\: :::))) for 483.2 In thk
combination of symbols only the additive principle is employed.
Somewhat later,3 10 is represented also by a horizontal dash; the
1 G. H. F. Ncsselmann, Die Algebra der Griechen (Berlin, 1842), p. 72, 494;
M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), P- 126, 127.
2 Arthur J. Evans, Scripla Minoa, Vol. I (1909), p. 258, 256.
8 Arthur J. Evans, The Palace of Minos (London, 1921), Vol. 1, p 646; see
also p. 279.
22 A HISTORY OF MATHEMATICAL NOTATIONS
sloping line indicative of 100 and the lozenge-shaped figure used for
1,000 were replaced by the forms O for 100, and <> for 1,000.
OOo o = = = I I I stood for 2>496 •
33. The oldest strictly Greek numeral symbols were the so-called
Hcrodianic signs, named after Herodianus, a Byzantine grammarian
of about 200 A.D., who describes them. These signs occur frequently
in Athenian inscriptions and are, on that account, now generally
called Attic. They were the initial letters of numeral adjectives.1
They were used as early as the time of Solon, about GOO B.C., and con-
tinued in use for several centuries, traces of them being found as late
as the time of Cicero. From about 470 to 350 B.C. this system existed
in competition with a newer one to be described presently. The
Herodianic signs were
1 Iota for 1 II Eta for 100
II or TI or F Pi for 5 X Chi for 1,000
A Delta for 10 M My for 10,000
34. Combinations of the symbols for 5 with the symbols for 10,100,
1,000 yielded symbols for 50, 500, 5,000. These signs appear on an
abacus found in 1847, represented upon a Greek marble monument on
the island of Salamis.2 This computing table is represented in Fig-
ure 12.
The four right-hand signs I C T X, appearing on the horizontal
line below, stand for the fractions -J, ^, -£±, 4*8, respectively. Proceed-
ing next from right to left, we have the symbols for 1, 5, 10, 50, 100,
500, 1,000, 5,000, and finally the sign T for 6,000. The group of sym-
bols drawn on the left margin, and that drawn above, do not contain
the two symbols for 5,000 and 6,000. The pebbles in the columns
represent the number 9,823. The four columns represented by the
five vertical lines on the right were used for the representation of the
fractional values J, -^-5-, ^J4, 4J, respectively.
35. Figure 13 shows the old Herodianic numerals in an Athenian
state record of the fifth century B.C. The last two lines are: Ke0<xA(uoj>
1 See, for instance. G. Friedlein, Die Zahlzeichen und das elementarc Rechnen dcr
Griechen und Romer (Krlangen, 1869), p. 8; M. Cantor, Vorlesungen uber Geschichte
der Mathemaiik, Vol. I (3d ed.), p. 120; II. Ilankel, Zur Geschichte der Mathemalik
im Alter thum und Mittelalter (Leipzig, 1874), p. 37.
2 Kubitschek, "Die Salaminische Rechentafel," Numismatische Zeitschrift
(Vienna, 1900), Vol. XXXI, p. 393-98; A. Nagl, ibid., Vol. XXXV (1903), p. 131-
43; M. Cantor, Kulturleben der Volker (Halle, 1863), p. 132, 136; M. Cantor, Vor-
lesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 133.
OLD NUMERAL SYMBOLS
23
ai/a[Xcoarosr] oD eiri r[r?s] apxw HHHPTTT....; i.e., "Total
of expenditures during our office three hundred and fifty-three
talents "
36. The exact reason for the displacement of the Herodianic sym-
bols by others is not known. It has been suggested that the com-
mercial intercourse of Greeks with the Phoenicians, Syrians, and
Hebrews brought about the change. The Phoenicians made one im-
portant contribution to civilization by their invention of the alpha-
bet. The Babylonians and Egyptians had used their symbols to
represent whole syllables or words. The Phoenicians borrowed hieratic
X
o
JL
<
a.
E.
x
TPXPHPAPHCTX
FIG. 12. — The computing table of Salamis
signs from Egypt and assigned them a more primitive function as
letters. But the Phoenicians did not use their alphabet for numerical
purposes. As previously seen, they represented numbers by vertical
and horizontal bars. The earliest use of an entire alphabet for desig-
nating numbers has been attributed to the Hebrews. As previously
noted, the Syrians had an alphabet representing numbers. The
Greeks are supposed by some to have copied the idea from the He-
brews. But Moritz Cantor1 argues that the Greek use is the older and
that the invention of alphabetic numerals must be ascribed to the
Greeks. They used the twenty-four letters of their alphabet, together
with three strange and antique letters, ST (old van), 9 (koppa), *)
(sampi), and the symbol M. This change was decidedly for the worse,
for the old Attic numerals were less burdensome on the memory inas-
1 V&rlesungen uber Geschichte der Mathematik, Vol. I (3d ed., 1907), p. 25.
24
A HISTORY OF MATHEMATICAL NOTATIONS
FIG. 13. — Account of disbursements of the Athenian state, 418-415 B.C.,
British Museum, Greek Inscription No. 23. (Taken from R. Brown, A History of
Accounting and Accountants [Edinburgh, 1905], p. 26.)
OLD NUMERAL SYMBOLS 25
much as they contained fewer symbols. The following are the Greek
alphabetic numerals and their respective values:
aftyde^frjBi K X/i v £ o TT 9
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
P <r r v <p x t « ^ ,a ,/3 ,7,
100 200 300 400 500 600 700 800 900 1,000 2,000 3,000
etc.
P v
M M M, etc.
10,000 20,000 30,000
37. A horizontal line drawn over a number served to distinguish
it more readily from words. The coefficient for M was sometimes
placed before or behind instead of over the M. Thus 43,678 was
written SM^x^- The horizontal line over the Greek numerals
can hardly be considered an essential part of the notation ; it does not
seem to have been used except in manuscripts of the Byzantine
period.1 For 10,000 or myriad one finds frequently the symbol M or
Mu, sometimes simply the dot • , as in /3-o5 for 20,074. Often2 the
coefficient of the myriad is found written above the symbol /iu.
38. The paradox recurs, Why did the Greeks change from the
Herodianic to the alphabet number system? Such a change would
not be made if the new did not seem to offer some advantages over the
old. And, indeed, in the new system numbers could be written in a
more compact form. The Herodianic representation of 1,739 was
X HlHIIAAAII MM; the alphabetic was ,a^X0. A scribe might consider
the latter a great innovation. The computer derived little aid from
either. Some advantage lay, however, on the side of the Herodianic,
as Cantor pointed out. Consider HHIIH+HH= SI H,AAAA+AA = S|A;
there is an analogy here in the addition of hundred's and of ten's.
But no such analogy presents itself in the alphabetic numerals, where
the corresponding steps are v+a = x and /Z+K = £; adding the hun-
dred's expressed in the newer notation affords no clew as to the sum
of the corresponding ten's. But there was another still more impor-
tant consideration which placed the Herodianic far above the alpha-
betical numerals. The former had only six symbols, yet they afforded
an easy representation of numbers below 100,000; the latter demanded
twenty-seven symbols for numbers below 1,000! The mental effort
1 Encyc. des stien. math., Tome I, Vol. I (1904), p. 12. 2 Ibid.
26 A HISTORY OF MATHEMATICAL NOTATIONS
of remembering such an array of signs was comparatively great. We
are reminded of the centipede having so many legs that it could
hardly advance.
39. We have here an instructive illustration of the fact that a
mathematical topic may have an amount of symbolism that is a hin-
drance rather than a help, that becomes burdensome, that obstructs
progress. We have here an early exhibition of the truth that the move-
ments of science are not always in a forward direction. Had the Greeks
not possessed an abacus and a finger symbolism, by the aid of which
computations could be carried out independently of the numeral
notation in vogue, their accomplishment in arithmetic and algebra
might have been less than it actually was.
40. Notwithstanding the defects of the Greek system of numeral
notation, its use is occasionally encountered long after far better
systems were generally known. A Calabrian monk by the name of
Barlaam,1 of the early part of the fourteenth century, wrote several
mathematical books in Greek, including arithmetical proofs of the
second book of Euclid's Elements, and six books of Logistic, printed in
1564 at Strassburg arid in several later editions. In the Logistic he de-
velops the computation with integers, ordinary fractions, and sexa-
gesimal fractions; numbers are expressed by Greek letters. The
appearance of an arithmetical book using the Greek numerals at as
late a period as the close of the sixteenth century in the cities of Strass-
burg and Paris is indeed surprising.
41. Greek writers often express fractional values in words. Thus
Archimedes says that the length of a circle amounts to three diameters
and a part of one, the size of which lies between one-seventh and ten-
seventy-firsts.2 Eratosthenes expresses J J of a unit arc of the earth's
meridian by stating that the distance in question "amounts to eleven
parts of which the meridian has eighty-three. "3 When expressed in
symbols, fractions were often denoted by first writing the numerator
marked with an accent, then the denominator marked with two ac-
cents and written twice. Thus,4 if KCL" KCL" = |f. Archimedes, Euto-
cius, and Diophantus place the denominator in the position of the
1 All our information on Barlaam is drawn from M. Cantor, Vorlesungen liber
Geschichte der Matkematik, Vol. I (3d ed.), p. 509, 510; A. G. Kastner, Geschichte der
Mathematik (Gottingen, 1796), Vol. I, p. 45; J. C. Hcilbronner, Historia matheseos
universae (Lipsiae, 1742), p. 488, 489.
2 Archimedis opera omnia (ed. Heiberg; Leipzig, 1880), Vol. I, p. 262.
3 Ptolemaus, MeyaXij avvrafa (ed. Heiberg), Pars I, Lib. 1, Cap. 12, p. 68.
4 Heron, Stereometrica (ed. Hultsch; Berlin, 1864), Pars I, Par. 8, p. 155.
OLD NUMERAL SYMBOLS 27
modern exponent; thus1 Archimedes and Eutocius use the notation
__ KO! Ka
if or if for ^], and Diophantus (§§ 101-6), in expressing large num-
bers, writes (Ariihmetica, Vol. IV, p. 17), — — ^ for -w-L.-,- .
7-/^X/ca 2,704
Here the sign ~ takes the place of the accent. Greek writers, even as
late as the Middle Ages, display a preference for unit fractions, which
played a dominating role in old Egyptian arithmetic.2 In expressing
such fractions, the Greeks omitted the a for the numerator and wrote
the denominator only once. Thus ju6//=4V- Unit fractions in juxta-
position were added,3 as in f" /cr?" pt/3" o-/c6// = ^+^V+iH+ T*4- ^ne
finds also a single accent,4 as in 5' = \. Frequent use of unit fractions is
found in Gcminus (first century B.C.), Diophantus (third century A.D.),
Eutocius and Proclus (fifth century A.D.). The fraction \ had a mark
of its own,5 namely, L or £, but this designation was no more
adopted generally among the Greeks than were the other notations
of fractions. Ptolemy6 wrote 38°50' (i.e., 380,i |) thus, XT;' £'7'".
Hultsch has found in manuscripts other symbols for |, namely, the
semicircles £VI, (, and the sign ,S ; the origin of the latter is uncertain.
He found also a symbol for §, resembling somewhat the small omega
(co).7 Whether these symbols represent late practice, but not early
usage, it is difficult to determine with certainty.
42. A table for reducing certain ordinary fractions to the sum of
unit fractions is found in a Greek papyrus from Egypt, described by
1 G. II. F. Nessclmarm, Algebra der Gricchcn (Berlin, 1842), p. 114.
2 ,1. Baillct describes a papyrus, "Le papyrus mathematique d'Akhmfm," in
Memoires publics par Ics immbrcs de la Mission archeologique fran^aise au Caire
(Paris, 1892), Vol. IX, p. 1-89 (8 plates). This papyrus, found at Akhmtrn, in
Egypt, is written in Greek, and is supposed to belong to the period between 500 and
800 A.D. It contains a table for the conversion of ordinary fractions into unit frac-
tions.
3Fr. Hultsch, Metrologicorum scriplorum reliquiae (1864-66), p. 173-75; M.
Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.), p. 129.
4 Nesselmann, op. cil., p. 112.
5 Ibid.; James Gow, Short History of Greek Mathematics (Cambridge, 1884),
p. 48, 50.
*Geographia (ed. Carolus Mullerus; Paris, 1883), Vol. I, Part I, p. 151.
7 Metrologicorum scriptorum reliquiae (Leipzig, 1804), Vol. I, p. 173, 174. On
p. 175 and 176 Hultsch collects the numeral symbols found in three Parisian manu-
scripts, written in Greek, which exhibit minute variations in the symbolism. For
instance, 700 is found to be ^ ^, \j/f.
28 A HISTORY OF MATHEMATICAL NOTATIONS
L. C. Karpinski,1 and supposed to be intermediate between the
Ahmes papyrus and the Akhmim papyrus. Karpinski (p. 22) says:
"In the table no distinction is made between integers and the corre-
sponding unit fractions; thus 7' may represent either 3 or £, and
actually y'y' in the table represents 3^. Commonly the letters used
as numerals were distinguished in early Greek manuscripts by a bar
placed above the letters but not in this manuscript nor in the Akhmim
papyrus.7' In a third document dealing with unit fractions, a Byzan-
tine table of fractions, described by Herbert Thompson,2 f is written
1; i, a; J, f (from \ '); |, A/" (from A'); *, e (from e'); i, vf (from
H'). As late as the fourteenth century, Nicolas Rhabdas of Smyrna
wrote two letters in the Greek language, on arithmetic, containing
tables for unit fractions.3 Here letters of the Greek alphabet used as
integral numbers have bars placed above them.
43. About the second century before Christ the Babylonian sexa-
gesimal numbers were in use in Greek astronomy; the letter omicron,
which closely resembles in form our modern zero, was used to desig-
nate a vacant space in the writing of numbers. The Byzantines wrote
it usually b, the bar indicating a numeral significance as it has when
placed over the ordinary Greek letters used as numerals.4
44. The division of the circle into 360 equal parts is found in
Hypsicles.5 Hipparchus employed sexagesimal fractions regularly, as
did also C. Ptolemy6 who, in his Almagest, took the approximate
8 30
value of TT to be 3+^+™^-^ • ^n ^ne Heiberg edition this value is
written 7 rj X, purely a notation of position. In the tables, as printed
by Heiberg, the dash over the letters expressing numbers is omitted.
In the edition of N. Halma7 is given the notation 7 ?/ X", which is
1 "The Michigan Mathematical Papyrus No. 621," Isis, Vol. V (1922), p.
20-25.
2 "A Byzantine Table of Fractions," Ancient Egypt, Vol. I (1914), p. 52-54.
3 The letters were edited by Paul Tannery in Notices et extraits des manuscrits
de la Bibliotheque Nationale, Vol. XXXII, Part 1 (1886), p. 121-252.
4 C. Ptolemy, Almagest (ed. N. Halma; Paris, 1813), Book I, chap, ix, p. 38
and later; J. L. Heiberg, in his edition of the Almagest (Syntaxis mathematical
(Leipzig, 1898; 2d ed., Leipzig, 1903), Book I, does not write the bar over the o
but places it over all the significant Greek numerals. This procedure has the ad-
vantage of distinguishing between the o which stands for 70 and the o which stands
for zero. See Encyc. des scien. math., Tome I, Vol. I (1904), p. 17, n. 89.
5 Ava<£optKos (ed. K. Manitius), p. xxvi.
6 Syntaxis mathematica (ed. Heiberg), Vol. I, Part 1, p. 513.
7 Composition math, de PtoUmee (Paris, 1813), Vol. I, p. 421; see also Encyc. des
scien. math., Tome I, Vol. I (1904), p. 53, n. 181.
OLD NUMERAL SYMBOLS
29
probably the older form. Sexagesimal fractions were used during the
whole of the Middle Ages in India, and in Arabic and Christian coun-
tries. One encounters them again in the sixteenth and seventeenth
centuries. Not only sexagesimal fractions, but also the sexagesimal
notation of integers, are explained by John Wallis in his Maihesis
universalis (Oxford, 1657), page 68, and by V. Wing in his Astronomia
Briiannica (London, 1652, 1669), Book I.
EARLY ARABS
45. At the time of Mohammed the Arabs had a script which did
not differ materially from that of later centuries. The letters of the
early Arabic alphabet came to be used as numerals among the Arabs
1 t
5
6
20 «:
so <!
40 r
50 c
60
70
80
90
^
100 d
.200 ^
300^
400 c»
500 &
600 £
700 o
800 o»
900 Jb
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000 j,
20000 £J
30000 jj
40000 £o
50000 jj
60000 ^
70000 .«*•
80000 «
90000 «
100000 g
200000 £,
300000 jxi
400000 43
500000 J3
600000 j>
700000 ^
800000 -£* i
900000 «
FIG. 14. — Arabic alphabetic numerals used before the introduction of the
Hindu-Arabic numerals.
as early as the sixth century of our era.1 After the time of Mohammed,
the conquering Moslem armies coming in contact with Greek culture
acquired the Greek numerals. Administrators and military leaders
used them. A tax record of the eighth century contains numbers
expressed by Arabic letters and also by Greek letters.2 Figure 14 is
a table given by Ruska, exhibiting the Arabic letters and the numerical
values which they represent. Taking the symbol for 1,000 twice, on
the multiplicative principle, yielded 1,000,000. The Hindu-Arabic
1 Julius Ruska, "Zur altesten arabischen Algebra und Rechenkunst," Sitzungs-
berichte d. Heidelberger Akademie der Wissensch. (Philos.-histor. Klasse, 1917; 2.
Abhandlung), p. 37.
2 Ibid., p. 40.
30 A HISTORY OF MATHEMATICAL NOTATIONS
numerals, with the zero, began to spread among the Arabs in the nin
and tenth centuries, and they slowly displaced the Arabic and Gre<
numerals.1
ROMANS
46. We possess little definite information on the origin of tl
Roman notation of numbers. The Romans never used the successr
letters of their alphabet for numeral purposes in the manner practic<
by the Syrians, Hebrews, and Greeks, although (as we shall see) *
alphabet system was at one time proposed by a late Roman write
Before the ascendancy of Rome the Etruscans, who inhabited ti
country nearly corresponding to modern Tuscany and who ruled
Rome until about 500 B.C., used numeral signs which resembled lette
of their alphabet and also resembled the numeral signs used by tl
Romans. Moritz Cantor2 gives the Etrurian and the old Roman sign
as follows: For 5, the Etrurian /\ or V, the old Roman V; for 10 tl
Etrurian X or +, the old Roman X; for 50 the Etrurian t or I, tl
old Roman "f or I or X or 1 or L; for 100 the Etrurian 0, the o
Roman © ; for 1,000 the Etrurian #, the old Roman 0. The reser
blance of the Etrurian numerals to Etrurian letters of the alphabet
seen from the following letters: V, +, I, O, 8. These resemblanc
cannot be pronounced accidental. "Accidental, on the other hand
says Cantor, "appears the relationship with the later Roman signs,
V, X, L, C, M, which from their resemblance to letters transformc
themselves by popular etymology into these very letters/7 The origii
of the Roman symbols for 100 and 1,000 are uncertain; those for '
and 500 are generally admitted to be the result of a bisection of tl
two former. "There was close at hand/' says G. Friedlein,3 "the a
breviation of the word centum and mille which at an early age brougl
about for 100 the sign C, and for 1,000 the sign M and after Augustu
M." A view held by some Latinists6 is that "the signs for 50, 10
1,000 were originally the three Greek aspirate letters which the R«
mans did not require, viz., M>, O, 0, i.e., x> 0> *• The *& was writte
J_ and abbreviated into L; O from a false notion of its origin made HI
1 Ibid., p. 47.
2 Vorlesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 523, and t'
table at the end of the volume.
3 Die Zahlzeichen und das elementare Rechnen der Griechen und Homer (E
langen, 1869), p. 28.
4 Theodor Mommsen, Die unleritalischen Dialekte (Leipzig, 1840), p. 30.
'Ritschl, Rhein. Mus., Vol. XXIV (1869), p. 12.
OLD NUMERAL SYMBOLS 31
the initial of centum; and 0 assimilated to ordinary letters CIO.
The half of 0, viz., D, was taken to be -J- 1,000, i.e., 500; X probably
from the ancient form of 0, viz., ®, being adopted for 10, the half
of it V was taken for 5."1
47. Our lack of positive information on the origin and early his-
tory of the Roman numerals is not due to a failure to advance working
hypotheses. In fact, the imagination of historians has been unusually
active in this field.2 The dominating feature in the Roman notation is
the principle of addition, as seen in II, XII, CC, MDC, etc.
48. Conspicuous also is the frequent use of the principle of sub-
traction. If a letter is placed before another of greater value, its
value is to be subtracted from that of the greater. One sees this in
IV, IX, XL. Occasionally one encounters this principle in the Baby-
lonian notations. Remarks on the use of it are made by Adriano
Cappelli in the following passage :
"The well-known rule that a smaller number, placed to the left
of a larger, shall be subtracted from the latter, as 0|00 = 4,000, etc.,
was seldom applied by the old Romans and during the entire Middle
Ages one finds only a few instances of it. The cases that I have found
belong to the middle of the fifteenth century and are all cases of IX,
never of IV, and occurring more especially in French and Piedmontese
documents. Walther, in his Lexicon diptomaticum, Gottingen, 1745-
47, finds the notation LXL = 90 in use in the eighth century. On the
other hand one finds, conversely, the numbers IIIX, VIX with the
meaning of 13 and 16, in order to conserve, as Lupi remarks, the Latin
terms tertio dedmo and sexto decimo."* L. C. Karpinski points out
that the subtractive principle is found on some early tombstones and
on a signboard of 130 B.C., where at the crowded end of a line 83 is
written XXCIII, instead of LXXXIII.
1 II. J. Roby, A Grammar of the Latin Language from Plaulus to Suetonius
(4th ed.; London, 1881), Vol. I, p. 441.
2 Consult, for example, Friedlcin, op. cit., p. 26-31; Ncsselmann, op. tit.,
p. 86-92; Cantor, Mathematische Beitrdge zum Kulturleben der Volker, p. 155-67;
J. C. Heilbronner, Historia Matheseos universae (Lipsiae, 1742), p. 732-35; Grotc-
fend, Lateinische Grammatik (3d ed.; Frankfurt, 1820), Vol. II, p. 163, is quoted in
the article "Zahlzeichen" in G. S. Kliigel's Malhematisches Worterbuch, continued
by C. B. Mollweide and J. A. Grunert (Leipzig, 1831); Mommsen, Hermes, Vol.
XXII (1887), p. 596; Vol. XXIII (1888), p. 152. A recent discussion of the history
of the Roman numerals is found in an article by Ettore Bortolotti in Bolletino delta
Mathesis (Pavia,. 1918), p. 60-66, which is rich in bibliographical references, as is
also an article by David Eugene Smith in Scientia (July- August, 1926).
3 Lexicon Abbreviaturarum (Leipzig, 1901), p. xlix.
32 A HISTORY OF MATHEMATICAL NOTATIONS
49. Alexander von Humboldt1 makes the following observations:
"Summations by juxtaposition one finds everywhere among the
Etruscans, Romans, Mexicans and Egyptians; subtraction or lessen-
ing forms of speech in Sanskrit among the Indians: in 19 or unavinsati;
99 unusata; among the Romans in undeviginti for 19 (unus de viginti),
undeoctoginta for 79; duo de quadraginta for 38; among the Greeks
tikosi deonta henos 19, and pentekonta duoin deontoin 48, i.e., 2 missing
in 50. This lessening form of speech has passed over in the graphics of
numbers when the group signs for 5, 10 and even their multiples, for
example, 50 or 100, are placed to the left of the characters they modify
(IV and IA, XL and XT for 4 and 40) among the Romans and Etrus-
cans (Otfried Miiller, Etrusker, II, 317-20), although among the latter,
according to Otfried Miiller's new researches, the numerals descended
probably entirely from the alphabet. In rare Roman inscriptions
which Marini has collected (Iscrizioni della Villa di Albano, p. 193;
Hervas, Aritmetica delle nazioni [1786], p. 11, 16), one finds even 4
units placed before 10, for example, IIIIX for 6."
50. There are also sporadic occurrences in the Roman nota-
tions of the principle of multiplication, according to which VM
does not stand for 1,000 — 5, but for 5,000. Thus, in Pliny's His-
toria naturalis (about 77 A.D.), VII, 26; XXXIII, 3; IV praef., one
finds2 LXXXIII.M, XCII.M, CX.M for 83,000, 92,000, 110,000,
respectively.
51. The thousand-fold value of a number was indicated in some
instances by a horizontal line placed above it. Thus, Aelius Lam-
pridius (fourth century A.D.) says in one place, "CXX, equitum Persa-
rum fudimus: et mox X in bello interemimus," where the numbers
designate 120,000 and 10,000. Strokes placed on top and also on the
sides indicated hundred thousands; e.g., |X|CLXXXDC stood for
1,180,600. In more recent practice the strokes sometimes occur only
on the sides, as in | X | • DC . XC . , the date on the title-page of Sigii-
enza's Libra astronomicaj published in the city of Mexico in 1690.
In antiquity, to prevent fraudulent alterations, XXXM was written
for 30,000, and later still CIO took the place of M.3 According to
1 "liber die bei verschiedenen Volkern ublichen Systeme von Zahlzeichen,
etc./' Crclle's Journal fur die reine und angewandte Mathematik (Berlin, 1829),
Vol. IV, p. 210, 211.
2 Nesselmann, op. cit., p. 90.
3 Confer, on this point, Theodor Mommsen and J. Marquardt, Manuel des
antiquites romaines (trans. G. Humbert), Vol. X by J. Marquardt (trans. A. Vigie";
Paris, 1888), p. 47, 49.
OLD NUMERAL SYMBOLS 33
Cappelli1 "one finds, often in French documents of the Middle Ages,
the multiplication of 20 expressed by two small x's which are placed
as exponents to the numerals III, VI, VIII, etc., as in IIIIXX = 80,
VIXXXI = 131."
52. A Spanish writer2 quotes from a manuscript for the year 1392
the following:
M C
"IIII, IIII, LXXIII florins" for 4,473 florins.
M XX
"III C IIII III florins" for 3,183 (?) florins.
In a Dutch arithmetic, printed in 1771, one finds8
c c m c
t ffitj for 123, i j£ttj ittj toj for 123,456.
53. For 1,000 the Romans had not only the symbol M, but also I,
oo and CIO. According to Priscian, the celebrated Latin grammarian
of about 500 A.D., the oo was the ancient Greek sign X for 1,000, but
modified by connecting the sides by curved lines so as to distinguish it
from the Roman X for 10. As late as 1593 the oo is used by C. Dasypo-
dius4 the designer of the famous clock in the cathedral at Strasbourg.
The CIO was a I inclosed in parentheses (or apostrophes). When only
the right-hand parenthesis is written, 10, the value represented is
only half, i.e., 500. According to Priscian,5 "quinque milia per I et
duas in dextera parte apostrophes, 100- decem milia per supra dictam
formam additis in sinistra parte contrariis duabus notis quam sunt
apostrophi, CCIOO." Accordingly, 100 stood for 5,000, CCIOO for
10,000; also 1000 represented 50,000; and CCCIOOO, 100,000;
(co), 1,000,000. If we may trust Priscian, the symbols that look like
the letters C, or those letters facing in the opposite direction, were
not really letters C, but were apostrophes or what we have called
1 Op. cit., p. xlix.
2 Liciniano Saez, Demostracidn Histdrica del verdadero valor de *Todas Las
Monedas que corrlan en Castilla durante el reynado del Senor Don Enrique III
(Madrid, 1796).
3 De Vernieuwde Cyfferinge van Mf Willem B art j ens. Herstelt, .... door
Mr Jan van Dam, .... en van alle voorgaande Fauten gezuyvert door ....
Klaas Bosch (Amsterdam, 1771), p. 8.
4 Cunradi Dasypodii Institutionum Mathematicarum voluminis primi Erotemata
(1593), p. 23. .
6 "De figuris numerorum," Henrici Keilii Grammaiid Latini (Lipsiae, 1859),
Vol. Ill, 2, p. 407.
34 A HISTORY OF MATHEMATICAL NOTATIONS
parentheses. Through Priscian it is established that this notation is
at least as old as 500 A.D. ; probably it was much older, but it was not
widely used before the Middle Ages.
54. While the Hindu- Arabic numerals became generally known
in Europe about 1275, the Roman numerals continued to hold a com-
manding place. For example, the fourteenth-century banking-house
of Peruzzi in Florence — Compagnia Peruzzi — did not use Arabic
numerals in their account-books. Roman numerals were used, but
the larger amounts, the thousands of lira, were written out in words;
one finds, for instance, "Ib. quindicimilia CXV / V ^ VI in fiorini"
for 15,115 lira 5 soldi 6 denari; the specification being made that the
lira are lira a fiorino cVoro at 20 soldi and 12 denari. There appears
also a symbol much like ? , for thousand.1
Nagl states also: "Specially characteristic is .... during all the
Middle Ages, the regular prolongation of the last I in the units, as
VI |= VI I, which had no other purpose than to prevent the subsequent
addition of a further unit/'
55. In a book by H. Giraua Tarragones2 at Milan the Roman
numerals appear in the running text and are usually underlined; in
the title-page, the date has the horizontal line above the numerals.
The Roman four is 1 1 1 1 . In the tables, columns of degrees and minutes
are headed "G.M."; of hour and minutes, "H.M." In the tables, the
Hindu-Arabic numerals appear ; the five is printed 3 , without the
usual upper stroke. The vitality of the Roman notation is illustrated
further by a German writer, Sebastian Frank, of the sixteenth cen-
tury, who uses Roman numerals in numbering the folios of his book
and in his statistics: "Zimmet kuinpt von Zailon .CC.VN LX.
teiitscher meil von Calicut weyter gelegen ..... Die Nagelin kummen
von Meluza / fur Calicut hinaussgelegen vij-c. vnd XL. deutscher
meyl."3 The two numbers given are 260 and 740 German miles. Pe-
culiar is the insertion of vnd ("and")- Observe also the use of the
principle of multiplication in vij«c. ( = 700). In Jakob Kobel's
Rechenbiechlin (Augsburg, 1514), fractions appear in Roman numerals;
11°
thus, c~ Stands f°r *^*
1 Alfred Nagl, Zeitschrift fur Mathematik und Physik, Vol. XXXIV (1889),
Historisch-literarische Abthcilung, p. 164.
2 Dos Libros de Cosmographie, complicates nueuamcnte por Hieronymo
Giraua Tarragones (Milan, M.D.LVI).
8 Weltbuch I siriegel vnd bildtnis des gantzen Erdtbodens .... von Sebastiano
Franco W&rdensi ____ (M.D. XXXIIII), fol. ccxx.
OLD NUMERAL SYMBOLS
35
56. In certain sixteenth-century Portuguese manuscripts on navi-
gation one finds the small letter b used for 5, and the capital letter R
for 40. Thus, z&iij stands for 18, Rii] for 43.1
to
FIG. 15. — Degenerate forms of Roman numerals in English archives (Common
Pleas, Plea Rolls, 637, 701, and 817; also Recovery Roll 1). (Reduced.)
A curious development found in the archives of one or two English
courts of the fifteenth and sixteenth centuries2 was a special Roman
1 J. I. de Brito Rcbcllo, Livro de Marinharia (Lisboa, 1903), p. 37, 85-91, 193,
194.
2 Antiquaries Journal (London, 1926), Vol. VI, p. 273, 274.
36
A HISTORY OF MATHEMATICAL NOTATIONS
numeration for the membranes of their Rolls, the numerals assuming
a degraded form which in its later stages is practically unreadable.
In Figure 15 the first three forms show the number 147 as it was
written in the years 1421, 1436, and 1466; the fourth form shows the
number 47 as it was written in 1583.
57. At the present time the Roman notation is still widely used in
marking the faces of watches and clocks, in marking the dates of
books on title-pages, in numbering chapters of books, and on other
occasions calling for a double numeration in which confusion might
arise from the use of the same set of numerals for both. Often the
Roman numerals are employed for aesthetic reasons.
58. A striking feature in Roman arithmetic is the partiality for
duodecimal fractions. Why duodecimals and not decimals? We can
only guess at the answer. In everyday affairs the division of units
into two, three, four, and six equal parts is the commonest, and
duodecimal fractions give easier expressions for these parts. Nothing
definite is known regarding the time and place or the manner of the
origin of these fractions. Unlike the Greeks, the Romans dealt with
concrete fractions. The Roman as, originally a copper coin weighing
one pound, was divided into 12 unciae. The abstract fraction \-\- was
called deuna ( = de unaa, i.e., as [1] less uncia [r2]). Each duodecimal
subdivision had its own name and symbol. This is shown in the follow-
ing table, taken from Friedlein,1 in which S stands for semis or "half"
of an as.
TABLE
as
1
n
f
t
I7*
5
T2
i 1
1
SK.
deunx
S r = - or S : : •
S = = or S : :
S = - or S r 1 or & : •
£- or _ £_ or £:
$ — or 6Y<
S
r — — or — — — or : : •
— — or X S or : :
— — or — 1 or :•
— or z or :
-LL-Ii't
- or • or on bronze abacus (
:>ccur also curved ones /^/.
(de uncia 1— -fy)
f(de sextans 1~J)
\ (decem unciae)
(de quadrans 1— J)
(duae assis sc. partes)
(scptem unciae)
dextans 1
(decunx)J
dodrans
bes
septunx
semis
quincunx
(quinque unciae)
triens . ...
quadrans . .
sextans . . .
sescuncia 1J
uncia
^
In place of straight lines - <
1 Op. cit., Plate 2, No. 13; see also p. 35.
OLD NUMERAL SYMBOLS 37
59. Not all of these names and signs were used to the same ex-
tent. Since i+i=f, there was used in ordinary life | and £ (semis et
triens) in place of $ or \\ (decunx). Nor did the Romans confine them-
selves to the duodecimal fractions or their simplified equivalents
1; i> l> 1> etc., but used, for instance, TV in measuring silver, a libella
being TV denarius. The uncia was divided in 4 siciliciy and in 24 scripuli
etc.1 In the Geometry of Boethius the Roman symbols are omitted
and letters of the alphabet are used to represent fractions. Very
probably this part of the book is not due to Boethius, but is an inter-
polation by a writer of later date.
60. There are indeed indications that the Romans on rare occa-
sions used letters for the expression of integral numbers.2 Theodor
Mommsen and others discovered in manuscripts found in Bern,
Einsiedeln, and Vienna instances of numbers denoted by letters.
Tartaglia gives in his General trattato di nvmeri, Part I (1556), folios 4,
5, the following:
A 500 II R 80
B 300 K 51 S 70
C 100 L 50 T 160
D 500 M 1,000 V 5
E 250 N 90 X 10
F 40 0 11 Y 150
G 400 P 400 Z 2,000
H 200 Q 500
61. Gerbert (Pope Sylvestre II) and his pupils explained the Ro-
man fractions. As reproduced by Olleris,3 Gerbert's symbol for \
does not resemble the capital letter $, but rather the small letter <J .
1 For additional details and some other symbols used by the Romans, consult
Friedlein, p. 33-46 and Plate 3; also H. Hankel, op. tit., p. 57-61, where com-
putations with fractions are explained. Consult also Fr. Hultsch, Metrologic.
scriplorcs Romani (Leipzig, 1866).
2 Friedlein, op. tit., p. 20, 21, who gives references. In the Standard Dic-
tionary of the English Language (New York, 1896), under S, it is stated that 3
stood for 7 or 70.
1 (Euvres de Gerbert (Paris, 1867), p. 343-48, 393-96, 583, 584.
38 A HISTORY OF MATHEMATICAL NOTATIONS
PERUVIAN AND NORTH AMERICAN KNOT RECORDS1
ANCIENT QUIPU
62. 'The use of knots in cords for the purpose of reckoning, and
recording numbers" was practiced by the Chinese and some other
ancient people; it had a most remarkable development among the
Inca of Peru, in South America, who inhabited a territory as large as
the United States east of the Rocky Mountains, and were a people of
superior mentality. The period of Inca supremacy extended from
about the eleventh century A.D. to the time of the Spanish conquest
in the sixteenth century. The quipu was a twisted woolen cord, upon
which other smaller cords of different colors were tied. The color,
length, and number of knots on them and the distance of one from
another all had their significance. Specimens of these ancient quipu
have been dug from graves.
63. We reproduce from a work by L. Leland Locke a photograph of
one of the most highly developed quipu, along with a line diagram of
the two right-hand groups of strands. In each group the top strand
usually gives the sum of the numbers on the four pendent strands.
Thus in the last group, the four hanging strands indicate the numbers
89, 258, 273, 38, respectively. Their sum is 658; it is recorded by the
top string. The repetition of units is usually expressed by a long knot
formed by tying the overhand knot and passing the cord through the
loop of the knot as many times as there are units to be denoted. The
numbers were expressed on the decimal plan, but the quipu were not
adopted for calculation; pebbles and grains of maize were used in com-
puting.
64. Nordenskiold shows that, in Peru, 7 was a magic number; for
in some quipu, the sums of numbers on cords of the same color, or
the numbers emerging from certain other combinations, are multiples
of 7 or yield groups of figures, such as 2777, 777, etc. The quipu dis-
close also astronomical knowledge of the Peruvian Indians.2
65. Dr. Leslie Spier, of the University of Washington, sends me the
following facts relating to Indians in North America: "The data that
I have on the quipu-\ike string records of North-American Indians
indicate that there are two types. One is a long cord with knots and
1 The data on Peru knot records given here are drawn from a most interesting
work, The Ancient Quipu or Peruvian Knot Record, by L. Leland Locke (American
Museum of Natural History, 1923). Our photographs are from the frontispiece
and from the diagram facing p. 16. See Figs. 16 and 17.
2Erland Nordenskiold, Comparative Ethnographical Studies, No. 6, Part 1
(1925), p. 36.
OLD NUMERAL SYMBOLS
39
bearing beads, etc., to indicate the days. It is simply a string record.
This is known from the Yakima of eastern Washington and some In-
terior Salish group of Nicola Valley,1 B.C.
FIG. 16. — A quipu, from ancient Chancay in Peru, now kept in the American
Museum of Natural History (Museum No. B8713) in New York City.
1 J. D. Leechman and M. R. Harrington, String Records of the Northwest,
Indian Notes and Monographs (1921).
40
A HISTORY OF MATHEMATICAL NOTATIONS
"The other type I have seen in use among the Havasupai and
Walapai of Arizona. This is a cord bearing a number of knots to indi-
cate the days until a ceremony, etc. This is sent with the messenger
who carries the invitation. A knot is cut off or untied for each day that
elapses; the last one indicating the night of the dance. This is also
used by the Northern and South-
ern Maidu and the Miwok of Cali-
fornia.1 There is a mythical ref-
erence to these among the Zufii
of New Mexico.2 There is a note
on its appearance in San Juan
Pueblo in the same state in the
seventeenth century, which would
indicate that its use Was widely
known among the Pueblo Indians.
'They directed him (the leader of
the Pueblo rebellion of 1680) to
make a rope of the palm leaf and
tie in it a number of knots to rep-
resent the number of days be-
fore the rebellion was to take
place; that he must send the
rope to all the Pueblos in the
Kingdom, when each should sig-
nify its approval of, and union
with, the conspiracy by untying
one of the knots/3 The Huichol
of Central Mexico also have knot-
ted strings to keep count of days,
untieing them as the days elapse.
They also keep records of their lovers in the same way.4 The Zufii
also keep records of days worked in this fashion.6
1 R. B. Dixon, "The Northern Maidu," Bulletin of the American Museum of
Natural History, Vol. XVII (1905), p. 228, 271 ; P.-L. Faye, "Notes on the Southern
Maidu," University of California Publications of American Archaeology and
Ethnology, Vol. XX (1923), p. 44; Stephen Powers, "Tribes of California/' Contri-
butions to North American Ethnology, Vol. Ill (1877), p. 352.
2 F. H. Gushing, "Zufti Breadstuff," Indian Notes and Monographs, Vol. VIII
(1920), p. 77.
3 Quoted in J. G. Bourke, "Medicine-Men of the Apache," Ninth Annual
Report, Bureau of American Ethnology (1892), p. 555.
4 K. Lumholtz, Unknown Mexico, Vol. II, p. 218-30.
6 Leechman and Harrington, op. cit.
<**
FIG. 17. — Diagram of the two right-
hand groups of strands in Fig. 16.
OLD NUMERAL SYMBOLS 41
"Bourke1 refers to medicine cords with olivella shells attached
among the Tonto and Chiricahua Apache of Arizona and the Zufii.
This may be a related form.
"I think that there can be no question the instances of the second
type are historically related. Whether the Yakima and Nicola Valley
usage is connected with these is not established. "
AZTECS
66. "For figures, one of the numerical signs was the dot (•), which
marked the units, and which was repeated either up to 20 or up to the
figure 10, represented by a lozenge. The number 2Q was represented
by a flag, which, repeated five times, gave the number 100, which was
.:: O P
Xi^Jto 10 &* 6 fVO 100 3t>0
$jjk *§s£ || IP IP u I . "I " 1 1 1 I
FIG. 18. — Aztec numerals
marked by drawing quarter of the barbs of a feather. Half the barbs
was equivalent to 200, three-fourths to 300, the entire feather to 400.
Four hundred multiplied by the figure 20 gave 8,000, which had a
purse for its symbol."2 The symbols were as shown in the first line of
Figure 18.
The symbols for 20, 400, and 8,000 disclose the number 20 as the
base of Aztec numeration; in the juxtaposition of symbols the additive
principle is employed. This is seen in the second line3 of Figure 18,
which represents
2X8,000+400+3X20+3X5+3 = 16,478 .
67. The number systems of the Indian tribes of North America,
while disclosing no use of a symbol for zero nor of the principle of
1 Op. cit.y p. 550 ff.
2 Lucien Biart, The Aztecs (trans. 3. L. Garner; Chicago, 1905), p. 319.
8 Consult A. F. Pott, Die quindre und vigesimale Zdhlmethode bei Volkern aller
Welttheile (Halle, 1847).
42
A HISTORY OF MATHEMATICAL NOTATIONS
»***.*»*» +***
" '
FIG. 19. — From the Dresden Codex, of the Maya, displaying numbers. The
second column on the left, from above down, displays the numbers 9, 9, 16, 0, 0,
which stand for 9X 144,000+9X7,200+16 X360-fO+0 = 1,366,560. In the third
column are the numerals 9, 9, 9, 16, 0, representing 1,364,360. The original appears
in black and red colors. (Taken from Morley, An Introduction to the Study of the
Maya Hieroglyphs, p. 266.)
OLD NUMERAL SYMBOLS 43
local value, are of interest as exhibiting not only quinary, decimal, and
vigesimal systems, but also ternary, quaternary, and octonary sys-
tems.1
MAYA
68. The Maya of Central America and Southern Mexico developed
hieroglyphic writing, as found on inscriptions and codices, dating
apparently from about the beginning of the Christian Era, which dis-
closes the use of a remarkable number system and chronology.2
The number system discloses the application of the principle of local
value, and the use of a symbol for zero centuries before the Hindus
began to use their symbol for zero. The Maya system was vigesimal,
except in one step. That is, 20 units (kins, or "days") make 1 unit of
the next higher order (uinals, or 20 days), 18 uinals make 1 unit of the
third order (tun, or 360 days), 20 tuns make 1 unit of the fourth order
(Katun, or 7,200 days), 20 Katuns make 1 unit of the fifth order (cycle,
or 144,000 days), and finally 20 cycles make 1 great cycle of 2,880,000
days. In the Maya codices we find symbols for 1-19, expressed by
bars and dots. Each bar stands for 5 units, each dot for 1 unit. For
instance,
•• — ^- = .
1245 7 11 19
The zero is represented by a symbol that looks roughly like a half-
closed eye. In writing 20 the principle of local value enters. It is
expressed by a dot placed over the symbol for zero. The numbers are
written vertically, the lowest order being assigned the lowest position
(see Fig. 19). The largest number found in the codices is 12,489,781.
CHINA AND JAPAN
69. According to tradition, the oldest Chinese representation of
number was by the aid of knots in strings, such as are found later
among the early inhabitants of Peru. There are extant two Chinese
tablets3 exhibiting knots representing numbers, odd numbers being
designated by white knots (standing for the complete, as day, warmth,
1 W. C. Eells, "Number-Systems of North-American Indians,7' American
Mathematical Monthly, Vol. XX (1913), p. 263-72, 293-99; also Bibliotheca mathe-
matica (3d series, 1913), Vol. XIII, p. 218-22.
2 Our information is drawn from S. G. Morley, An Introduction to the Study of
the Maya Hieroglyphs (Washington, 1915).
3 Paul Perrty, Grammaire de la langue chinoise orale et ecrite (Paris, 1876),
Vol. II, p. 5-7; Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.),
p. 674.
44
A HISTORY OF MATHEMATICAL NOTATIONS
the sun) while even numbers are designated by black knots (standing
for the incomplete, as night, cold, water, earth). The left-hand tablet
shown in Figure 20 represents the numbers 1-10. The right-hand
tablet pictures the magic square of nine cells in which the sum of each
row, column, and diagonal is 15.
70. The Chinese are known to have used three other systems of
writing numbers, the Old Chinese numerals, the mercantile numerals,
and what have been designated as scientific numerals. The time of the
introduction of each of these systems is uncertain.
o 6 o
o
FIG. 20. — Early Chinese knots in strings, representing numerals
71. The Old Chinese numerals were written vertically, from above
down. Figure 21 shows the Old Chinese numerals and mercantile
numerals, also the Japanese cursive numerals.1
72. The Chinese scientific numerals are made up of vertical and
horizontal rods according to the following plan : The numbers 1-9 are
represented by the rods |, ||, |||, ||||, |||||, J, JL IJi Iffi; the numbers
10-90 are written thus _ = = = = ._!_=]= = ==• According to the
Chinese author Sun-Tsu, units are represented, as just shown, by
vertical rods, ten's by horizontal rods, hundred's again by vertical
rods, and so on. For example, the number 6,728 was designated by
73. The Japanese make use of the Old Chinese numerals, but have
two series of names for the numeral symbols, one indigenous, the other
derived from the Chinese language, as seen in Figure 21.
1 See also Ed. Biot, Journal asiatique (December, 1839), p. 497-502; Cantor,
Vorlesungen uber Geschichte der Maihematik, Vol. I, p. 673; Biernatzki, Crelle's
Journal, Vol. LII (1856), p. 59-94.
HINDU-ARABIC NUMERALS
HINDU-ARABIC NUMERALS
45
74. Introduction. — It is impossible to reproduce here all the forms
of our numerals which have been collected from sources antedating
1500 or 1510 A.D. G. F. Hill, of the British Museum, has devoted a
CHINOIS
CH1FFKES
VALEUBS.
NOMS DE NOMBttE
JAPOIfAIS
cursifs.
DU
COMMERCE.
EN >
JAP01UIS PUR.
Elf
SINICO-JAPOBTAIS.
—
—
1
1
ftots.
itsi.
^
^
f(
»
foutats.
ni.
H
^5.
W
3
mils.
san.
E3
w^3
^
l\
. yots.
si.
3L
rfS
^r
5
itsouts.
g*
-^
^
_t.
6
mouts.
rok.
-t
-t
j.
7
nanats.
silsi.
A
A
±
8
yatfi.
fats.
A
^l
3
9
kolconots.
kou.
-f*
-f
t
10
towd.
zyou.
lif
"S
^
100
moino.
fakoufyak.
•=f*
^
f
1,000
teidzi.
sen.
M
*
^
10,000
yorodz.
man.
Fie. 21. — Chinese and Japanese numerals. (Taken from A. P. Pihan, Expose
des signes de numeration [Paris, 1860], p. 15.)
whole book1 of 125 pages to the early numerals in Europe alone. Yet
even Hill feels constrained to remark: "What is now offered, in the
shape of just 1,000 classified examples, is nothing more than a vinde-
1 The Development of Arabic Numerals in Europe (exhibited in 64 tables;
Oxford, 1915).
46 A HISTORY OF MATHEMATICAL NOTATIONS
miatio prima" Add to the Hill collection the numeral forms, or sup-
posedly numeral forms, gathered from other than European sources,
and the material would fill a volume very much larger than that of
Hill. We are compelled, therefore, to confine ourselves to a few of the
more important and interesting forms of our numerals.1
75. One feels the more inclined to insert here only a few tables of
numeral forms because the detailed and minute study of these forms
has thus far been somewhat barren of positive results. With all the
painstaking study which has been given to the history of our numerals
we are at the present time obliged to admit that we have not even
settled the time and place of their origin. At the beginning of the
present century the Hindu origin of our numerals was supposed to
have been established beyond reasonable doubt. But at the present
time several earnest students of this perplexing question have ex-
pressed grave doubts on this point. Three investigators — G. II. Kaye
in India, Carra de Vaux in France, and Nicol. Bubnov in Russia —
working independently of one another, have denied the Hindu origin.2
However, their arguments are far from conclusive, and the hypothesis
of the Hindu origin of our numerals seems to the present writer to
explain the known facts more satisfactorily than any of the substitute
hypotheses thus far advanced.3
1 The reader who desires fuller information will consult Hill's book which is
very rich in bibliographical references, or David Eugene Smith and Louis Charles
Karpinski's The Hindu-Arabic Numerals (Boston and London, 1911). See also an
article on numerals in English archives by H. Jenkinson in Antiquaries Journal,
Vol. VI (1926), p. 263-75. The valuable original researches due to F. Woepcke
should be consulted, particularly his great "Mdmoirc sur la propagation des
chiffres indiens" published in the Journal asiatique (6th series; Paris, 1863), p. 27-
79, 234-90, 442-529. Reference should be made also to a few other publications of
older date, such as G. Friedlein's Zahlzcichen und das elementare Rechnen der
Griechen und Homer (Erlangen, 1869), which touches questions relating to our
numerals. The reader will consult with profit the well-known histories of mathe-
matics by H. Hankel and by Moritz Cantor.
2 G. R. Kaye, "Notes on Indian Mathematics," Journal and Proceedings of the
Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 475-508; "The Use of the Abacus
in Ancient India," ibid., Vol. IV (1908), p. 293-97; "References to Indian Mathe-
matics in Certain Mediaeval Works," ibid., Vol. VII (1911), p. 801-13; "A Brief
Bibliography of Hindu Mathematics," ibid., p. 679-86; Scientia, Vol. XXIV
(1918), p. 54; "Influence grecque dans le de"veloppement des mathc'matiques
hindoues," ibid., Vol. XXV (1919), p. 1-14; Carra de Vaux, "Sur 1'origine des
chiffres," ibid., Vol. XXI (1917), p. 273-82; Nicol. Bubnov, Arithmetische Selbst-
stdndigkeit der europdischen Kultur (Berlin, 1914) (trans, from Russian cd.; Kiev,
1908).
3F. Cajori, "The Controversy on the Origin of Our Numerals," Scientific
Monthly, Vol. IX (1919), p. 458-64. See also B. Da'tta in Amer. Math. Monthly,
Vol. XXXIII, p. 449; Proceed. Benares Math. Soc.t Vol. VII.
HINDU-ARABIC NUMERALS 47
76. Early Hindu mathematicians, Aryabhata (b. 476 A.D.) and
Brahmagupta (b. 598 A.D.), do not give the expected information
about the Hindu-Arabic numerals.
Aryabhata's work, called Aryabhatiya, is composed of three parts,
in only the first of which use is made of a special notation of numbers.
It is an alphabetical system1 in which the twenty-five consonants
represent 1-25, respectively; other letters stand for 30, 40, . . . . ,
100, etc.2 The other mathematical parts of Aryabhata consists of
rules without examples. Another alphabetic system prevailed in
Southern India, the numbers 1-19 being designated by consonants,
etc.3
In Brahmagupta's Pulverizer, as translated into English by H. T.
Colebrooke,4 numbers are written in our notation with a zero and the
principle of local value. But the manuscript of Brahmagupta used by
Colebrooke belongs to a late century. The earliest commentary on
Brahmagupta belongs to the tenth century; Colebrooke's text is
later.5 Hence this manuscript cannot be accepted as evidence that
Brahmagupta himself used the zero and the principle of local value.
77. Nor do inscriptions, coins, and other manuscripts throw light
on the origin of our numerals. Of the old notations the most impor-
tant is the Brahmi notation which did not observe place value and in
which 1, 2, and 3 are represented by , , = . The forms of the
Brahmi numbers do not resemble the forms in early place-value nota-
tions6 of the Hindu-Arabic numerals.
Still earlier is the Kharoshthi script,7 used about the beginning of
the Christian Era in Northwest India and Central Asia. In it the first
three numbers are I II III, then X = 4, IX = 5, IIX = 6, XX = 8, 1 = 10,
3 = 20, 33=40, 133 = 50, XI =100. The writing proceeds from right
to left.
78. Principle of local value. — Until recently the preponderance of
authority favored the hypothesis that our numeral system, with its
concept of local value and our symbol for zero, was wholly of Hindu
origin. But it is now conclusively established that the principle of
1 M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), p.
606.
2 G. R. Kaye, Indian Mathematics (Calcutta and Simla, 1915), p. 30, gives
full explanation of Aryabhata's notation.
8 M. Cantor, Math. Beitrdge z. Kulturkben der Volkcr (1863), p. 68, 69.
4 Algebra with Arithmetic and Mensuration from the Sanscrit (London, 1817),
p. 326 ff.
6 Ibid., p. v, xxxii.
6 See forms given by G. R. Kaye, op. cit., p. 29. 7 Ibid.
48 A HISTORY OF MATHEMATICAL NOTATIONS
local value was used by the Babylonians much earlier than by the
Hindus, and that the Maya of Central America used this principle
and symbols for zero in a well-developed numeral system of their
own and at a period antedating the Hindu use of the zero (§ 68).
79. The earliest-known reference to Hindu numerals outside of
India is the one due to Bishop Severus Sebokht of Nisibis, who, living
in the convent of Kenneshre on the Euphrates, refers to them in a
fragment of a manuscript (MS Syriac [Paris], No. 346) of the year
662 A.D. Whether the numerals referred to are the ancestors of the
modern numerals, and whether his Hindu numerals embodied the
principle of local value, cannot at present be determined. Apparently
hurt by the arrogance of certain Greek scholars who disparaged the
Syrians, Sebokht, in the course of his remarks on astronomy and
mathematics, refers to the Hindus, " their valuable methods, of cal-
culation ; andjjieir computing that surpasses description. Ijwish only
to say that this computation is done by means of nine signs."1
80. Some interest attaches to the earliest dates indicating the use
of the perfected Hindu numerals. That some kind of numerals with a
earlier than the ninth century is indicated by
Brahmagupta (b. 598 A.D.), who gives rules for computing with a
#ero.2 G. Biihler3 believes he has found definite mention of the decimal
system and zero m the year 620 A.D. These statements do not neces-
sarily imply the use of a decimal" system based on the principle of
local value. G. R. Kaye4 points out that the task of the antiquarian is
complicated by the existence of forgeries. In the eleventh century in
India "there occurred a specially great opportunity to regain con-
fiscated endowments and to acquire fresh ones." Of seventeen cita-
tions of inscriptions before the tenth century displaying the use of
place value in writing numbers, all but two are eliminated as forgeries;
these two are for the years 813 and 867 A.D.; Kaye is not sure of the
reliability even of these. According to D.JE. Smith _and JLjg.^ Kar-
pinski,5 the earliest authentic document unmistakably containing the
numerals mttMyh^^r^njMia belongs to the year 876 A.D. The earli-
1 See M. F. Nau, Journal asiatique (10th ser., 1910), Vol. XVI, p. 255; L. C.
Karpinski, Science, Vol. XXXV (1912), p. 969-70; J. Ginsburg, Bulletin of the
American Mathematical Society, Vol. XXIII (1917), p. 368.
2 Colebrooke, op. cit.r p. 339, 340.
3 "Indische Palaographie," Grundriss d. indogerman. Philologie u. Alieriuma-
kunde, Band I, Heft 11 (Strassburg, 1896), p. 78.
4 Journal of the Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 482-87.
5 The Hindu-Arabic Numerals (New York, 1911), p. 52.
HINDU-ARABIC NUMERALS
49
est Arabic manuscripts containing the numerals are c>f_874l and 888
A.D. They appear again in a work written at Shiraz in Persia2 in 970 A.D.
A church pillar3 not far from the Jeremias Monastery in Egypt has
I
a
3
4
5
6
7
8
9
10
it
la
'3
I
T
I
I
I
1
i
t
*
7
I
t
I
1
CD
r
IT
"6
T
T
t
S
er
r
z
&
r
M
5
Ih
rt
u
B
fifi
V
*
Q
b
1?
S
H
O
*r
b
b
K
b
t
b
L
T
V
/t
A
V
V
V
v
V
A
X
A
yy,
8
3
a
8
8
sr
8
8
6
8
B
&
8
8
8
I
x
S
2
/
6
CO
5
S>
976
x
1077
Ixi
XI
XI or XII
| beg.
XII?
XII
XII*
C. 1200
C. 1200
XII
XV
XVI early
FIG. 22. — G. F. Hill's table of early European forms and Boethian apices.
(From G. F. Hill, The Development of Arabic Numerals in Europe [Oxford, 1915],
p. 28. Mr. Hill gives the MSS from which the various sets of numerals in this table
are derived: [1] Codex Vigilanus; [2] St. Gall MS now in Zurich; [3] Vatican MS
3101, etc. The Roman figures in the last column indicate centuries.)
1 Karabacek, Wiener Zeitschriftfur die Kunde des Morgenlandcs, Vol. II (1897),
p. 56.
2 L. C. Karpinski, Bibliotheca mathemalica (3d ser., 1910-11), p. 122.
3 Smith and Karpinski, op. cit., p. 138-43.
£ut Ant*.
NommliofMui-
50 A HISTORY OF MATHEMATICAL NOTATIONS
the date 349 A.H. ( = 961 A.D.). The oldest definitely dated European
manuscript known to contain the Hindu-Arabic numerals is the Codex
Vigilanus (see Fig. 22, No. 1), written in the Albelda Cloister in Spain
in 976 A.D. The nine characters without the zero are given, as an
addition, in a Spanish copy of the Origines by Isidorus of Seville,
992 A.D. A tenth-century manuscript with forms differing materially
from those in the Codex Vigilanus was found in the St. Gall manu-
script (see Fig. 22, No. 2), now in the University Library at Zurich.
[The numerals are contained in a Vatican manuscript of 1077 (see Fig.
22, No. 3), on a Sicilian coin of 1138, in a Regensburg (Bavaria)
•ssi^r £$^uuM>f*n
ApfOM Of Boothia* _. x» ^> r A. x->
*B4 of tho MUdlo | "£* ^k f\ f9 Cl [3 ^/\ g 5 ®
Sr.-r-1 iT7*9-cfrt/'?8cJ>«
Nam.rU, ofthoj p JXfjiortfoOT/Jn V A <\ •
I l>^}afiJVA3°
t^^tf'H^fcr^o
lyonthoJffmwr ^J 9 jf /"• x^A P Q C>
gSKtr- l 2 3 f y * \ * 3 °
JTSJir; i * 3~3*»4<i«5 6 A~7 890
WMHor(T).14«8. ^ v/^l|wv/\» v j ^
From />« ^«rl« -_ .g ^ x *%. > .
s«Pp«tfln<r« byi i. 54-5 6 78 ^10
TODlUU, 1M». ^J< /w/*^
FIG. 23.— Table of important numeral forms. (The first, six lines in this table
are copied from a table at the end of Cantor's Vorlesungen liber Geschichte der
Mathematik, Vol. 1. The numerals in the Bamberg arithmetic are taken from
Friedrich linger, Die Methodik der praktischen Arithmetik in historischer Eni-
wickelung [Leipzig, J88S], p. 39.)
chronicle of 1197. The earliest manuscript in French giving the
numerals dates about 1275. In the British Museum one English manu-
script is of about 1230-50; another is of 1246. The earliest undoubted
Hindu-Arabic numerals on a gravestone are at Pforzheim in Baden
of 1371 and one at Ulm of 1388. The earliest coins outside of Italy
that are dated in the Arabic numerals are as follows: Swiss 1424,
Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551.
81. Forms of numerals. — The Sanskrit letters of the second cen-
tury A.D. head the list of symbols in the table shown in Figure 23. The
implication is that the numerals have evolved from these letters. If
such a connection could be really established, the Hindu origin of our
numeral forms would be proved. However, a comparison of the forms
appearing in that table will convince most observers that an origin
HINDU-ARABIC NUMERALS 61
from Sanskrit letters cannot be successfully demonstrated in that
way; the resemblance is no closer than it is to many other alphabets.
The forms of the numerals varied considerably. The 5 was the
most freakish. An upright 7 was rare in the earlier centuries. The
symbol for zero first used by the Hindus was a dot.1 The symbol for
zero (0) of the twelfth anHTEIrteenth centuries i is ^sometiines crossed
b^ a horizontal line, or a line slanting upward.2 The Boethian apices,
as found in some manuscripts, contain a triangle inscribed in the
circular zero. In Athelard of Bath's translation of Al-Madjrltl's re-
vision of Al-Khowarizmi's astronomical tables there are in different
manuscripts three signs for zero,3 namely ,_JhejQ ( = theta?) referred
to above, then T ( = teca),* and 0. In oncToFlKe manuscripts 38 is
written severaTtlmes XXXO, and 28 is written XXO, the 0 being
intended most likely as the abbreviation for oclo ("eight")-
82. The symbol T for zero is found also in a twelfth-century
manuscript5 of N. Ocreatus, addressed to his master Athelard. In
that century it appears especially in astronomical tables as an ab-
breviation for tcca, which, as already noted, was one of several names
for zero;6 it is found in those tables by itself, without connection with
other numerals. The symbol occurs in the Alyorixmus vulyaris as-
cribed to Sacrobosco.7 C. A. Nallino found o for zero in a manuscript
of Escurial, used in the preparation of an edition of Al-Battani. The
[symbol © for zero occurs also in printed mathematical books.
The one author who in numerous writings habitually used 6 for
zero was the French mathematician Michael Rollc (1652-1719). One
finds it in his Traite d'algebre (1690) and in numerous articles in the
publications of the French Academy and in the Journal des s^avans.
1 Smith and Karpinski, op. cit., p. 52, 53.
2 Hill, op. cit., p. 30-60.
3 II. Suter, Die astronomischen Tafeln des Muhammed ibn Musd Al-Khwdrizml
in der Bearbeitang des Maslama ibn Ahmed Al-Madjrltl und der lateinischen Uebcr-
setzung des Alhelhard von Bath (K^benhavn, 1914), p. xxiii.
4 See also M. Curtze, Petri Philomeni de Dacia in Algorismum vulgarem
Johannis de Sacrobosco Commentarius (Hauniae, 1897), p. 2, 20.
6 "Prologus N. Ocreati in Helceph ad Adelardum Batcnscm Magistrum suum.
Fragment sur la multiplication et la division public* pour la premiere fois par
Charles Henry," Abhandlungen zur Geschichle der Mathematik, Vol. Ill (1880),
p. 135-38.
6 M. Curtze, Urkunden zur Geschichte der Mathematik im Mittelalter und der
Renaissance (Leipzig, 1902), p. 182.
7 M. Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig,
1898), p. 3-27.
52 A HISTORY OF MATHEMATICAL NOTATIONS
Manuscripts of the fifteenth century, on arithmetic, kept in the
Ashmolean Museum1 at Oxford, represent the zero by a circle, crossed
by a vertical stroke and resembling the Greek letter <£. Such forms
for zero arc reproduced by G. F. Hill2 in many of his tables of numer-
als.
83. In the fifty-six philosophical treatises of the brothers Ibwan
as-safa (about 1000 A.D.) are shown Hindu-Arabic numerals and the
corresponding Old Arabic numerals.
The forms of the Hindu-Arabic numerals, as given in Figure 24,
have maintained themselves in Syria to the present time. They ap-
pear with almost identical form in an Arabic school primer, printed
i«: ?: •* y
1: f r I
f> L: t: *
u 1 A v i 0 f r r t
Fir,. 24. — In the first line are the Old Arabic numerals for 10, 9, 8, 7, 6, 5, 4, 3,
2, 1. In the second line are the Arabic names of the numerals. In the third line
are the Hindu-Arabic numerals as given by the brothers Ifrwan as-safa. (Repro-
duced from J. Ruska, op. cit., p. 87.)
at Beirut (Syria) in 1920. The only variation is in the 4, which in 1920
assumes more the form of a small Greek epsilon. Observe that 0 is
represented by a dot, and 5 by a small circle. The forms used in mod-
ern Arabic schoolbooks cannot be recognized by one familiar only with
the forms used in Europe.
84. In fifteenth-century Byzantine manuscripts, now kept in the
Vienna Library,3 the numerals used are the Greek letters, but the
principle of local value is adopted. Zero is 7 or in some places • ; aa
means 11, py means 20, ayyy means 1,000. "This symbol 7 for zero
means elsewhere 5," says Heiberg, "conversely, o stands for 5 (as now
among the Turks) in Byzantine scholia to Euclid ..... In Constanti-
nople the new method was for a time practiced with the retention of
1 Robert Stcclc, The Earliest Arithmetics in English (Oxford, 1922), p. 5.
2 Op. cit., Tables III, IV, V, VI, VIII, IX, XI, XV, XVII, XX, XXI, XXII.
See also E. Wappler, Zur Geschichte der deutschen Algebra im XV. Jahrhundert
(Zwickauer Gymnasialprogramm von 1887), p. 11-30.
3 J. L. Heiberg, "Byzantinische Analekten," Abhandlungen zur Geschichte der
Mathematik, Vol. IX (Leipzig, 1899), p. 163, 166, 172. This manuscript in the
Vienna Library is marked "Codex Phil. Gr. 65."
HINDU-ARABIC NUMERALS 53
the old letter-numerals, mainly, no doubt, in daily intercourse." At
the close of one of the Byzantine manuscripts there is a table of
numerals containing an imitation of the Old Attic numerals. The table
gives also the Hindu-Arabic numerals, but apparently without recog-
nition of the principle of local value; in writing 80, the 0 is placed over
the 8. This procedure is probably due to the ignorance of the scribe.
85. A manuscript1 of the twelfth century, in Latin, contains the
symbol h for 3 which Curtze and Nagl2 declare to have been found
only in the twelfth century. According to Curtze, the foregoing
strange symbol for 3 is simply the symbol for tertia used in the nota-
tion for sexagesimal fractions which receive much attention in this
manuscript.
86. Recently the variations in form of our numerals have been sum-
marized as follows: "The form3 of the numerals 1, 6, 8 and 9 has not
varied much among the [medieval] Arabs nor among the Christians
of the Occident; the numerals of the Arabs of the Occident for 2, 3 and
5 have forms offering some analogy to ours (the 3 and 5 are originally
reversed, as well among the Christians as among the Arabs of the
Occident); but the form of 4 and that of 7 have greatly modified
themselves. The numerals 5, 6, 7, 8 of the Arabs of the Orient differ
distinctly from those of the Arabs of the Occident (Gobar numerals).
For five one still writes 5 and _J." The use of i for 1 occurs in the first
printed arithmetic (Treviso, 1478), presumably because in this early
. stage of printing there was no type for 1. Thus, 9,341 was printed
934;.
87. Many points of historical interest are contained in the fol-
lowing quotations from the writings of Alexander von Humboldt.
Although over a century old, they still are valuable.
"In the Gobar4 the group signs are dots, that is zeroes, for in
India, Tibet and Persia the zeroes and dots are identical. The Gobar
symbols, which since the year 1818 have commanded my whole at-
tention, were discovered by my friend and teacher, Mr. Silvestre de
Sacy, in a manuscript from the Library of the old Abbey St. Germain
du Pres. This great orientalist says: 'Le Gobar a un grand rapport
1 Algorithmus-MSS Clm 13021, fols. 27-29, of the Munich Staatsbibliothek.
Printed and explained by Maximilian Curtze, Abhandlungen zur Geschichte der
Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27.
2 Zeilschrift fur Mathematik und Physik (Hist. Litt. Abth.), Vol. XXXIV
(Leipzig, 1889), p. 134.
3 Encyc. des Stien. math., Tome I, Vol. I (1904), p. 20, n. 105, 106.
4 Alexander von Humboldt, Crelle's Journal, Vol. IV (1829), p. 223, 224.
54 A HISTORY OF MATHEMATICAL NOTATIONS
avec le chifYre indien, mais 11 n'a pas de z6ro (S. Gramm. arabe, p. 76,
and the note added to PL 8).' I am of the opinion that the zero-
symbol is present, but, as in the Scholia of Neophytos on the units, it
stands over the units, not by their side. Indeed it is these very zero-
symbols or dots, which give these characters the singular name Gobar
or dust-writing. At first sight one is uncertain whether one should
recognize therein a transition between numerals and letters of the
alphabet. One distinguishes with difficulty the Indian 3, 4, 5 and 9.
Dal and ha are perhaps ill-formed Indian numerals 6 and 2. The nota-
tion by dots is as follows:
3 ' for 30 ,
4" for 400,
6 •*• for 6,000 .
These dots remind one of an old-Greek but rare notation (Ducarige,
Palacogr., p. xii), which begins with the myriad: a" for 10,000, fl::
for 200 millions. In this system of geometric progressions a single dot,
which however is not written down, stands for 100. In Diophantus
and Pappus a dot is placed between letter-numerals, instead of the
initial Mv (myriad). A dot multiplies what lies to its left by 10,000.
.... A real zero symbol, standing for the absence of some unit, is ap-
plied by Ptolemy in the descending sexagesimal scale for missing de-
grees, minutes or seconds. Delambre claims to have found our sym-
bol for zero also in manuscripts of Theon, in the Commentary to the
Syntaxis of Ptolemy.1 It is therefore much older in the Occident than
the invasion of the Arabs and the work of Planudes on arithmoi
indikoi." L. C. Karpinski2 has called attention to a passage in the
Arabic biographical work, the Fihrist (987 A.D.), which describes a
Hindu notation using dots placed below the numerals; one dot indi-
cates tens, two dots hundreds, and three dots thousands.
88. There are indications that the magic power of the principle of
local value was not recognized in India from the beginning, and that
our perfected Hindu-Arabic notation resulted from gradual evolution.
Says Hurnboldt: "In favor of the successive perfecting of the designa-
tion of numbers in India testify the Tamul numerals which, by means
1 J. B. J. Delambre, Histoire de Vastron. ancienne, Vol. I, p. 547; Vol. II, p. 10.
The alleged passage in the manuscripts of Theon is not found in his printed works.
Delambre is inclined to ascribe the Greek sign for zero either as an abbreviation
of ouden or as due to the special relation of the numeral omicron to the sexagesimal
fractions (op. cit., Vol. II, p. 14, and Journal des sgavans [1817], p. 539).
*Bibliolheca malficmaiica, Vol. XI (1910-11), p. 121-24.
HINDU-ARABIC NUMERALS
55
of the nine signs for the units and by signs of the groups 10, 100, or
1,000, express all values through the aid of multipliers placed on the
r v +3 K^ :i5>ns-
xi, -X.jjL.^|.^|
^^^xiiffi
'i
>^-
mi%
•a
o
fl
o
a
H
left. This view is supported also by the singular arithmoi indikoi in
the scholium of the monk Neophytosy which is found in the Parisian
A HISTORY OF MATHEMATICAL NOTATIONS
Qit tJifler jiffir werben 0<»ontt<$ mit fren €&«*
ractcrn vie btrrwfc rdgulfo gift ribcn/|>abe flUlcfr*
volnirvil fonbcrc rcrn>anMun00r0cit ixngcmci*
fwnjiffcrn/aufttnomen *>A» f&nfft rn& ffoenfc.
4u$foU&u fbnberlufcnurcftn/ wenn bey cfncr
5i(f«r6rtypon<tOebn/fob«U5a(ftlbi0pag gcrrt fo
Ml £?mcr/»n& f tin turt^eil minbcr no$ mcbn
0ie balbcn 2?mcr tr<rt>en allein mit cincr Imi Oder
f!ri4>lm 9nrerr<beit>cn.9efi ale off t cin flricfclin btircfr
4<n jiffer gcbi/bcnlmpt re tin balben 2^mer/pnb ba«
<jef<fei^t tUtin (>cy 5<n iymcrn pnnb ntcfai t>ein pur/
*1
li<b9<ertcl!mebto^er
minber ober bit 0e4
funben £jrmcrbetr/
ba» wirbr bur^ die
jtxxy vo!0cnben jty<
n/ onb
'!'
fer b~tbtut\u vw .i^tfWeunbbalbfcymer.
• J O-f Jcbcntbalbtr
^ Btbeotbam'cr; --
\ ' let minder. JL* «»«!
Nf*w S
FIG. 26.— From Christoff Ru-
dolff's Kunstliche Rechnung mit der
Ziffer (Augsburg, 1574[?j).
Library (Cod. Reg., fol. 15), for an
account of which I am indebted to
Prof. Brandis. The nine digits of
Neophytos wholly resemble the Per-
sian, except the 4. The digits 1, 2,
3 and 9 are found even in Egyptian
number inscriptions (Koscgarten, de
Hierogl. AegypL, p. 54). The nine
units are enhanced tenfold, 100 fold,
1,000 fold by writing above them
one, two or three zeros, as in:
o o o o o°o
2-20, 24 = 24, 5 = 500, 6 = 6,000.
If we imagine dots in place of the
zero symbols, then we have the
arabic Gobar numerals."1 Humboldt
copies the scholium of Neophytos.
J. L. Heiberg also has called atten-
tion to the scholium of Neophytos
and to the numbering of scholia to
Euclid in a Greek manuscript of
the twelfth century (Codex Vindo-
bonensis, Gr. 103), in which numer-
als resembling the Gobar numerals
occur.2 The numerals of the monk
Neophytos (Fig. 25), of which
Humboldt speaks, have received the
special attention of P. Tannery.3
89. Freak forms. — We reproduce
herewith from the Augsburg edition
of Christoff Rudolff's Kunstliche
Rechnung a set of our numerals, and
of symbols to represent such fractions
1 Op. tit., p. 227.
2 See J. L. Heiberg's edition of Euclid
(Leipzig, 1888), Vol. V; P. Tannery, Revue
archeol. (3d scr., 1885), Vol. V, p. 99, also
(3d scr., 1886), Vol. VII, p. 355; Encyd
des scien. math., Tome I, Vol. I (1904),
p. 20, n. 102.
3 Memoir es scientifiques, Vol. IV (Tou-
louse and Paris, 1920), p. 22.
HINDU-ARABIC NUMERALS 57
and mixed numbers as were used in Vienna in the measurement of
wine. We have not seen the first edition (1526) of Rudolff's book,
but Alfred Nagl1 reproduces part of these numerals from the first
edition. "In the Viennese wine-cellars," says Hill, "the casks were
marked according to their contents with figures of the forms given."2
The symbols for fractions are very curious.
90. Negative numerals. — J. Colson3 in 1726 claimed that, by the
use of negative numerals, operations may be performed with "more
ease and expedition." If 8605729398715 is to be multiplied by
389175836438, reduce these to small numbers 1414331401315 and
4l 1224244442. Then write the multiplier on a slip of paper and
place it in an inverted position, so that its first figure is just over the
left-hand figure of the multiplicand. Multiply 4X1=4 and write
down 4. Move the multiplier a place to the right and collect the two
products, 4X1 + 1X1 = 5; write down 5. Move the multiplier another
place to the right, then 4X4+1X1 + 1X1 = 16; write the 1 in the
second line. Similarly, the next product is 11, and so on. Similar
processes and notations were proposed by A. Cauchy,4 E. Selling,5 and
W. B. Ford,6 while J. P. Ballantino7 suggests 1 inverted, thus i, as a
sign for negative 1, so that 1X7 = 13 and the logarithm 9 . 69897 - 10
may be written 19 . 69897 or I . 69897. Negative logarithmic charac-
teristics are often marked with a negative sign placed over the
numeral (Vol. II, §476).
91. Grouping digits in numeration. — In the writing of numbers con-
taining many digits it is desirable to have some symbol separating the
numbers into groups of, say, three digits. Dots, vertical bars, commas,
arcs, and colons occur most frequently as signs of separation.
In a manuscript, Liber algorizmi,8 of about 1200 A.D., there appear
1 Monatsblalt der numismatischen Gesellschaft in Wien, Vol. VII (December,
1906), p. 132.
2 G. F. Hill, op. cU., p. 53.
3 Philosophical Transactions, Vol. XXXIV (1726), p. 161-74; Abridged Trans-
actions, Vol. VI (1734), p. 2-4. See also G. Peano, Formulaire mathematique, Vol.
IV (1903), p. 49.
4 Comptcs rendus, Vol. XI (1840), p. 796; (Euvres (1st ser.), Vol. V, p. 434-55.
6 Eine mue Rechenrnaschine (Berlin, 1887), p. 16; see also Encyklopddie d.
Math. Wiss., Vol. I, Part 1 (Leipzig, 1898-1904), p. 944.
6 American Mathematical Monthly, Vol. XXXII (1925), p. 302.
7 Op. til., p. 302.
8M. Cantor, Zeitschrift fur Mathematik, Vol. X (1865), p. 3; G. Enestrom,
Bibliothcca mathematica (3d ser., 1912-13), Vol. XIII, p. 265.
58 A HISTORY OF MATHEMATICAL NOTATIONS
dots to mark periods of three. Leonardo of Pisa, in his Liber Abbaci
(1202), directs that the hundreds, hundred thousands, hundred mil-
lions, etc., be marked with an accent above; that the thousands,
millions, thousands of millions, etc., be marked with an accent below.
In the 1228 edition,1 Leonardo writes 678 935 784 105 296. Johannes
de Sacrobosco (d. 1256), in his Tractatus de arte numerandi, suggests
that every third digit be marked with a dot.2 His commentator,
Petrus de Dacia, in the first half of the fourteenth century, does the
same.3 Directions of the same sort are given by Paolo Dagomari4 of
Florence, in his Regoluzze di Maestro Paolo doll Abbaco and Paolo of
Pisa,5 both writers of the fourteenth century. Luca Pacioli, in his
Summa (1494), folio 196, writes 8 659 421 635 894 676; Georg Peur-
bach (1505),6 "3790528614. Adam Riese7 writes 86789325178. M.
Stifel (1544)8 writes 2329089562800. Gemma Frisius9 in 1540 wrote
24 456 345 678. Adam Riese (1535)10 writes 86 -7 -89 -3 -25 -178. The
Dutch writer, Martinus Carolus Creszfeldt,11 in 1557 gives in his
Arithmetica the following marking of a number:
"Exempei. || 5 8 7 4 9 3 6 2 5 3 4 || ."
w i w i w i w
1 El liber abbaci di Leonardo Pisano .... da B. Boncompagni (Roma, 1857),
p. 4.
2 J. O. Ilalliwcll, Rara malhematica (London, 1839), p. 5; M. Cantor, Vor-
lesungen, Vol. II (2d cd., 1913), p. 89.
3 Petri Philomeni de Dacia in Algorismum vulgar em lohannis de Sacrobosco
commentarius (ed. M. Curtze; Kopenhagen, 1897), p. 3, 29; J. Tropfke, Geschichte
der Nlemcntarmathematik (2d cd., 1921), Vol. I, p. 8.
4 Libri, Histoire des sciences mathematiques en Italic, Vol. Ill, p. 296-301
(Rule 1).
6 Ibid., Vol. II, p. 206, n. 5, and p. 526; Vol. Ill, p. 295; see also Cantor, op. cit.,
Vol. II (2d ed., 1913), p. 164.
6 Opus alyorithmi (Herbipoli, 1505). See Wildermuth, "Rechnen," Encyklo-
paedie des gesammten Erziehungs- und Unterrichtsivesens (Dr. K. A. Schmid, 1885).
7 Hechnung auff der Linien vnnd Federn (1544); Wildermuth, "Reehnen,"
Encijklopaedie (Schmid, 1885), p. 739.
8 Wildermuth, op. cit., p. 739.
9 Arithmetical practicae methodus facilis (1540) ; F. linger, Die Methodik der
praktischen Arithmetik in hislorischer Entwickelung (Leipzig, 1888), p. 25, 71.
10 Rechnung auff d. Linien u. Federn (1535). Taken from H. Hankel, op. cit.
(Leipzig, 1874), p. 15.
11 Arithmetica (1557). Taken from Bierens de Haan, Bouwstoffen voor de Ge-
schiedenis der Wis-en Natuurkundige Wetenschappent Vol. II (1887), p. 3.
HINDU-ARABIC NUMERALS 59
Thomas Blundeville (1636)1 writes 5|936|649. Tonstall2 writes
. ... 43210
3210987654321. Clavius3 writes 42329089562800. Chr. Rudolff4 writes
23405639567. Johann Caramuel6 separates the digits, as in "34:252,-
Integri. Partes.
341;154,329"; W. Oughtred,6 9!876i543|210l2i345678i9; K. Schott7,
7697432329089562436; N. Barreme,8 254.567.804.652; W. J. G.
Ill II I 0
Karsten,9872 094,826 152,870 364,008; I. A. de Segner,105|329//|870|
325/|743|297°, 174; Thomas Dilworth/1 789 789 789; Nicolas Pike,12
3 2 1
356;809,379;120,406;129,763; Charles Hutton,13 281,427,307; E.
Bczout,14 23, 456, 789, 234, 565, 456.
In M. Lcmos' Portuguese encyclopedia15 the population of New
1 Mr. Klundcvil, His Exercises contayning eight Treatises (7th cd., Ro. Hartwell;
London, 1636), p. 106.
2 De Artc Svppvtandi, libri qvatvor Cvtheberti Tonstalli (Argentorati), Colophon
1544, p. 5.
3 Christophori Clavii epitome arithmeticae practicac (Romae, 1583), p. 7.
4 Kunstliche Rechnung mil dcr Ziffer (Augsburg, 1574[?J), Aiij B.
6 Joannis Caramvclis mathesis biceps, veins et nova (Cornpaniae [southeast of
Naples], 1670), p. 7. The passage is as follows: "Punctum finale ( . ) est, quod poni-
tur post unitatem: ut cum scribirnus 23. viginti tria. Comma (,) post millemirium
scribitur . . . . ut cum scribimus, 23,424 Millcriarium & centenario -dis-
tinguere alios populos docent Hispani, qui utuntur hoc charactere \f , . . . . Hypo-
colon (;) millioncm a millcnario separat, ut cum scribimus 2;041,311. Duo puncta
ponuntur post billioncm, sen millioncm millionum, videlicet 34:252,341;154,329."
Caramuel was born in Madrid. For biographical sketch see Rcvista matemdtica
Hispano-American, Vol. I (1919), p. 121, 178, 203.
6 Clavis mathematicae (London, 1652), p. 1 (1st ed., 1631).
7 Cursus mathematicus (Herbipoli, 1661), p. 23.
8 Arithm6tique (new ed.; Paris, 1732), p. 6.
9 Mathesis theoretica elementaris atqve svblimior (Rostochii, 1760), p. 195.
10 Elementa arithmelicae gcomelriae et calcvli geometrici (2d ed.; Halle, 1767),
p. 13.
11 Schoolmaster's Assistant (22d ed.; London, 1784), p. 3.
12 New and Complete System of Arithmetic (Newburyport, 1788), p. 18.
18 " Numeration," Mathematical and Philosophical Dictionary (London, 1795).
14 Cours de malMmatiques (Paris, 1797), Vol. I, p. 6.
16 ' 'Portugal," Encyclopedia Portugueza ILlustrada . . . . de Maximiano Lemos
(Porto).
CO A HISTORY OF MATHEMATICAL NOTATIONS
York City is given as "3 .437:202"; in a recent Spanish encyclopedia,1
the population of America is put down as "150- 979,995."
In the process of extracting square root, two early commentators2
on Bhaskara's Lilavati, namely Rama-Crishna Deva and Gangad'hara
(ca. 1420 A.D.), divide numbers into periods of two digits in this man-
ner, 8 8 2 0 9. In finding cube roots Rama-Crishna Deva writes
i i — i
1953125.
92. The Spanish "calderon" — In Old Spanish and Portuguese
numeral notations there are some strange and curious symbols. In a
contract written in Mexico City in 1649 the symbols "7U291e" and
"VIIUCCXCIps" each represent 7,291 pesos. The U, which here re-
sembles an 0 that is open at the top, stands for "thousands."3 I. B.
Richman has seen Spanish manuscripts ranging from 1587 to about
1700, and Mexican manuscripts from 1768 to 1855, all containing
symbols for "thousands" resembling U or D, often crossed by one or
two horizontal or vertical bars. The writer has observed that after
1600 this U is used freely both with Hindu- Arabic and with Roman
numerals; before 1600 the U occurs more commonly with Roman
numerals. Karpinski has pointed out that it is used with the Hindu-
Arabic numerals as early as 1519, in the accounts of the Magellan
voyages. As the Roman notation does not involve the principle of
local value, U played in it a somewhat larger role than merely to
afford greater facility in the reading of numbers. Thus VIUCXV
equals 6X1,000+115. This use is shown in manuscripts from Peru
of 1549 and 1543,4 in manuscripts from Spain of 14805 and 1429.6
We have seen the corresponding type symbol for 1,000 in Juan Perez
de Moya,7 in accounts of the coming in the Real Casa de Moneda de
1 "America," Encyclopedia illmtrada segui Diccionario universal (Barcelona).
2 Colebrooke, op. cit., p. 9, 12, xxv, xxvii.
3 F. Cajori, "On the Spanish Symbol U for 'thousands/ " Bibliotheca mathe-
matica, Vol. XII (1912), p. 133.
4 Carlos de Indias publicalas por primer a vez d Ministerio de Fomento (Madrid,
1877), p. 502, 543, facsimiles X and Y.
5 Jose Gorizalo dc las Casas, Ancdes de la Palcoc/rafia Espanola (Madrid, 1857),
Plates 87, 92, 109, 110, 113, 137.
0 Liciniano Saez, Demoslracidn Histdricadel verdadero valor de todas las monedaa
que corrian en Caslilla duranle el Reynado del Senor Don Enrique III (Madrid,
1796), p. 447. See also Colomera y Rodriguez, Venancio, Paleoyrajia caslellana
(1862).
7 Arilmelica practica (14th ed.; Madrid, 1784), p. 13 (1st ed., 1562).
HINDU-ARABIC NUMERALS 61
Mexico (1787), in eighteenth-century books printed in Madrid,1
in the Gazetas de Mexico of 1784 (p. 1), and in modern reprints of
seventeenth-century documents.2 In these publications the printed
symbol resembles the Greek sampi 5 for 900, but it has no known
connection with it. In books printed in Madrid3 in 1760, 1655, and
1646, the symbol is a closer imitation of the written U, and is curiously
made up of the two /small printed letters, I, f, each turned halfway
around. The two inverted letters touch each other below, thus \f .
Printed symbols representing a distorted U have been found also in
some Spanish arithmetics of the sixteenth century, particularly in
that of Gaspard de Texeda4 who writes the number 103,075,102,300
in the Castellanean form c.iijU.75qs c.ijU300 and also in the algoristic
form 103U075qs 102U300". The Spaniards call this symbol and also
the sampi-like symbol a calderon.5 A non-Spanish author who ex-
plains the calderdn is Johann Caramuel,6 in 1670.
93. The present writer has been able to follow the trail of this
curious symbol U from Spain to Northwestern Italy. In Adriano
Cappelli's Lexicon is found the following: "In the liguric documents
of the second half of the fifteenth century we found in frequent use,
to indicate the multiplication by 1,000, in place of M, an O crossed
by a horizontal line."7 This closely resembles some forms of our
Spanish symbol U. Cappelli gives two facsimile reproductions8 in
1 Liciniano Saez, op. cit.
2 Manuel Danvila, Boletin de la Real Academia de la Hisloria (Madrid, 1888),
Vol. XII, p. 53.
3 Cuentas para lodas, compendia arilhmetico, e Histdrico . . . . su autor D.
Manuel Recio, Oficial de la contadurfa general de postos del Rcyno (Madrid, 1760) ;
Teatro Eclesidstico de la primitiva Iglesia de las Indias Occidentals .... el M. Gil
Gonzalez Davila, su Coronista Mayor de las Indias, y de los Reynos de las dos
Castillas (Madrid, 1655), Vol. II; Memorial, y Noticias Sacras, y reales del Imperio
de las Indias Occidentales .... Escriuiale por el afio de 1646, Juan Diez de la
Calle, Oficial Segundo de la Misma Secretaria.
*Suma de Arithmetica pratica (Valladolid, 1546), fol. iiijr.; taken from D. E.
Smith, History of Mathematics, Vol. II (1925), p. 88. The qs means quentos (cuen-
tos, "millions")-
5 In Joseph Aladern, Diccionari popular de la Llengua Catalana (Barcelona,
1905), we read under "Caldero": "Among ancient copyists a sign (\/") denoted
a thousand."
6 Joannis Caramvelis Mathesis biceps veins et nova (Companiae, 1670), p. 7.
7 Lexicon Abbreviaturarum (Leipzig, 1901), p. 1.
8 Ibid., p. 436, col. 1, Nos. 5 and 6.
62
A HISTORY OP MATHEMATICAL NOTATIONS
which the sign in question is small and is placed in the position of an
exponent to the letters XL, to represent the number 40,000. This
corresponds to the use of a small c which has been found written to the
right of and above the letters XI, to signify 1,100. It follows, there-
fore, that the modified U was in use during the fifteenth century in
Italy, as well as in Spain, though it is not known which country had
the priority.
What is the origin of this calderon? Our studies along this line
make it almost certain that it is a modification of one of the Roman
•F '• A
/* * !*'
FIG. 27. — From a contract (Mexi o City, 1649). The right part shows the sum
of 7,291 pesos, 4 tomines, 6 granos, ex >ressed in Roman numerals and the calderdn.
The left part, from the same contract, shows the same sum in Hindu-Arabic nu-
merals and the calderdn.
symbols for 1,000. Besides M, the Romans used for 1,000 the symbols
CIO, T, oo, and *f . These symbols are found also in Spanish manu-
scripts. It is easy to see how in the hands of successive generations of
amanuenses, some of these might assume the forms of the calderdn.
If the lower parts of the parentheses in the forms CIO or CIIO are
united, we have a close imitation of the U, crossed by one or by two
bars.
HINDU-ARABIC NUMERALS
63
94. The Portuguese "cifrao." — Allied to the distorted Spanish U is
the Portuguese symbol for 1,000, called the cifrao.1 It looks somewhat
like our modern dollar mark, $. But its function in writing numbers
was identical with that of the calderon. Moreover, we have seen forms
of this Spanish "thousand" which need only to be turned through a
right angle to appear like the Portuguese symbol for 1,000. Changes
of that sort are not unknown. For instance, the Arabic numeral 5
appears upside down in some Spanish books and manuscripts as late
as the eighteenth and nineteenth centuries.
a
FIG. 28. — Real estate sale in Mexico City, 1718. The sum written here is
4,255 pesos.
95. Relative size of numerals in tables. — Andr6 says on this point:
"In certain numerical tables, as those of Schron, all numerals are of
the same height. In certain other tables, as those of Lalande, of Cal-
let, of Houel, of Dupuis, they have unequal heights: the 7 and 9 are
prolonged downward; 3, 4, 5, 6 and 8 extend upward; while 1 and 2
do not reach above nor below the central body of the writing
The unequal numerals, by their very inequality, render the long
train of numerals easier to read; numerals of uniform height are less
legible."2
1 See the word cifrao in Antonio de Moraes Silva, Dice, de Lingua Portuguesa
(1877); in Vieira, Grande Dice. Portuguez (1873); in Dice. Comtemp. da Lingua
Portuguesa (1881).
2 D. Andre", Des notations math&matiques (Paris, 1909), p. 9.
64 A HISTORY OF MATHEMATICAL NOTATIONS
96. Fanciful hypotheses on the origin of the numeral forms. — A p
lem as fascinating as the puzzle of the origin of language relate
the evolution of the forms of our numerals. Proceeding on the t
assumption that each of our numerals contains within itself, {
skeleton so to speak, as many dots, strokes, or angles as it repres
units, imaginative writers of different countries and ages have
vanced hypotheses as to their origin. Nor did these writers feel i
they were indulging simply in pleasing pastime or merely contribu
to mathematical recreations. With perhaps only one exception, 1
were as convinced of the correctness of their explanations as are ch
squarers of the soundness of their quadratures.
The oldest theory relating to the forms of the numerals is du
the Arabic astrologer Aben Ragel1 of the tenth or eleventh cent
He held that a circle and two of its diameters contained the rcqu
forms as it were in a nutshell. A diameter represents 1; a diam
and the two terminal arcs on opposite sides furnished the 2. A glanc
Part I of Figure 29 reveals how each of the ten forms may be evol
from the fundamental figure.
On the European Continent, a hypothesis of the origin from do
the earliest. In the seventeenth century an Italian Jesuit wn
Mario Bettini,2 advanced such an explanation which was eag
accepted in 1651 by Georg Philipp Harsdorffer3 in Germany, '
said: "Some believe that the numerals arose from points or dots,
in Part II. The same idea was advanced much later by Geo
Dumesnil4 in the manner shown in the first line of Part III. In cur
writing the points supposedly came to be written as dashes, yielc
forms resembling those of the second line of Part III. The two hori:
tal dashes for 2 became connected by a slanting line yielding the n
ern form. In the same way the three horizontal dashes for 3 were joi
by two slanting lines. The 4, as first drawn, resembled the 0; but <
fusion was avoided by moving the upper horizontal stroke inl
1 J. F. Weidler, De characteribus numerorum vulgaribus dissertatio mathcma
critica (Wittembcrgae, 1737), p. 13; quoted from M. Cantor, Kulturleben der V\
(Halle, 1863), p. 60, 373.
2 Apiaria unwersae philosophiae, mathematicae, Vol. II (1642), Apiarium
p. 5. See Smith and Karpinski, op. cit., p. 36.
3 Delitae mathematicae et physicae (Niirnberg, 1651). Reference from
Sterner, Geschichte der Rechenkunst (Miinchen and Leipzig [1891]), p. 138, 52
4 "Note sur la forme des chifTres usuels," Revue archSologique (3d ser.; P
1890), Vol. XVI, p. 342-48. See also a critical article, "Pretendues notal
Pythagoriennes sur Forigine de nos chiffres," by Paul Tannery, in his Mem
scientifiques, Vol. V (1922), p. 8.
HINDU-ARABIC NUMERALS
65
vertical position and placing it below on the right. To avoid con-
founding the 5 and 6, the lower left-hand stroke of the first 5 was
fc7S?
3 D 5 E B
2 2 J 4 A 7 X
'i •'- H 5 G no %
s S a 5 B
I 2 3
o
8 O
S 88'S
o
s 2 O a
FIG. 29. — Fanciful hypotheses
<•
n
a
S "5
a o
changed from a vertical to a horizontal position and placed at the
top of the numeral. That all these changes were accepted as historical,
66 A HISTORY OF MATHEMATICAL NOTATIONS
without an atom of manuscript evidence to support the different steps
in the supposed evolution, is an indication that Baconian inductive
methods of research had not gripped the mind of Dumesnil. The origin
from dots appealed to him the more strongly because points played a
role in Pythagorean philosophy and he assumed that our numeral
system originated with the Pythagoreans.
Carlos le-Maur,1 of Madrid, in 1778 suggested that lines joining
the centers of circles (or pebbles), placed as shown in the first line of
Part IV, constituted the fundamental numeral forms. The explana-
tion is especially weak in accounting for the forms of the first three
numerals.
A French writer, P. Voizot,2 entertained the theory that originally
a numeral contained as many angles as it represents units, as seen in
Part V. He did not claim credit for this explanation, but ascribed it to
a writer in the Gcnova Catholico Militarite. But Voizot did originate
a theory of his own, based on the number of strokes, as shown in
Part VI.
Edouard Lucas3 entertains readers with a legend that Solomon's
ring contained a square and its diagonals, as shown in Part VII, from
which the numeral figures were obtained. Lucas may have taken this
explanation from Jacob Leupold4 who in 1727 gave it as widely current
in his day.
The historian Moritz Cantor5 tells of an attempt by Anton Miiller6
to explain the shapes of the digits by the number of strokes necessary
to construct the forms as seen in Part VIII. An eighteenth-century
writer, Georg Wachter,7 placed the strokes differently, somewhat as
in Part IX. Cantor tells also of another writer, Piccard,8 who at one
time had entertained the idea that the shapes were originally deter-
1 Elcmentos de Matematica pura (Madrid, 1778), Vol. I, chap. i.
J "Lcs chiffres arabes et leur origine," La nature (2d semestre, 1899), Vol.
XXVII, p. 222.
3 L' Arithmelique amusanle (Paris, 1895), p. 4. Also M. Cantor, Kulturlcben
der Volker (Halle, 1863), p. 60, 374, n. 116; P. Treutlcin, Geschichte unsercr Zahl-
zeicJien (Karlsruhe, 1875), p. 16.
4 Theatrvm Arithmetico-Geometricvm (Leipzig, 1727), p. 2 and Table III.
6 Kullurleben der Volker, p. 59, 60.
6 Arilhmetik und Algebra (Heidelberg, 1833). See also a reference to this in
P. Treutlein, op. tit. (1875), p. 15.
7 Naturae et Sctipturae Concordia (Lipsiae et Hafniae, 1752), chap. iv.
8 M6moire sur la forme et de la provenance des chiffres, Sociele Vaudoise des
sciences nalurelles (stances du 20 Avril et du 4 Mai, 1859), p. 176, 184. M. Cantor
reproduces the forms due to Piccard; see Cantor, Kidturleben, etc., Fig. 44.
HINDU-ARABIC NUMERALS 67
mined by the number of strokes, straight or curved, necessary to
express the units to be denoted. The detailed execution of this idea,
as shown in Part IX, is somewhat different from that of Mliller and
some others. But after critical examination of his hypothesis, Pic-
card candidly arrives at the conclusion that the resemblances he
pointed out are only accidental, especially in the case of 5, 7, and 9,
and that his hypothesis is not valid.
This same Piccard offered a special explanation of the forms of the
numerals as found in the geometry of Boethius and known as the
"Apices of Boethius." He tried to connect these forms with letters in
the Phoenician and Greek alphabets (see Part X). Another writer
whose explanation is not known to us was J. B. Reveillaud.1
The historian W. W. R. Ball2 in 1888 repeated with apparent ap-
proval the suggestion that the nine numerals were originally formed
by drawing as many strokes as there are units represented by the
respective numerals, with dotted lines added to indicate how the writ-
ing became cursive, as in Part XL Later Ball abandoned this ex-
planation. A slightly different attempt to build up numerals on the
consideration of the number of strokes is cited by W. Lietzmann.3
A still different combination of dashes, as seen in Part XII, was made
by the German, David Arnold Crusius, in 1746.4 Finally, C. P.
Sherman5 explains the origin by numbers of short straight lines, as
shown in Part XI11. "As time went on," he says, "writers tended
more and more to substitute the easy curve for the difficult straight
line and not to lift the pen from the paper between detached lines,
but to join the two — which we will call cursive writing."
These hypotheses of the origin of the forms of our numerals have
been barren of results. The value of any scientific hypothesis lies in
co-ordinating known facts and in suggesting new inquiries likely to
advance our knowledge of the subject under investigation. The hy-
potheses here described have done neither. They do not explain the
very great variety of forms which our numerals took at different times
1 Essai sur lea chiffrcs arabcs (Paris, 1883). Reference from Smith and Kar-
pinski, op. cit., p. 36.
2 A Short Account of the History of Mathematics (London, 1888), p. 147.
3 Lusliges und Merkwurdiges von Zahlen und Formcn (Brcslau, 1922), p. 73,
74. lie found the derivation in Raether, Theorie und Praxis dcs Rcchcnunterrichts
(1. Teil, 0. Aufl.; Brcslau, 1920), p. 1, who refers to H. von Jacobs, Das Volk der
Sicbener-Zdhler (Berlin, 1896).
4 Anweisung zur Rechen-Kunsl (Halle, 1746), p. 3.
6 Mathematics Teacher, Vol. XVI (1923), p. 398-401.
68 A HISTORY OF MATHEMATICAL NOTATIONS
and in different countries. They simply endeavor to explain the nu-
merals as they are printed in our modern European books. Nor have
they suggested any fruitful new inquiry. They serve merely as en-
tertaining illustrations of the operation of a pseudo-scientific imagina-
tion, uncontrolled by all the known facts.
97. A sporadic artificial system. — A most singular system of
numeral symbols was described by Agrippa von Nettesheim in his De
occulta philosophia (1531) and more fully by Jan Bronkhorst of Nim-
wegen in Holland who is named after his birthplace Noviomagus.1 In
1539 he published at Cologne a tract, De numeris, in which he de-
scribes numerals composed of straight lines or strokes which, he claims,
were used by Chaldaei et Astrologi. Who these Chaldeans are whom he
mentions it is difficult to ascertain; Cantor conjectures that they were
late Roman or medieval astrologers. The symbols are given again in
a document published by M. Host us in 1582 at Antwerp. An examina-
tion of the symbols indicates that they enable one to write numbers up
into the millions in a very concise form. But this conciseness is at-
tained at a great sacrifice of simplicity; the burden on the memory is
great. It does not appear as if these numerals grew by successive
steps of time; it is more likely that they are the product of some in-
ventor who hoped, perhaps, to see his symbols supersede the older
(to him) crude and clumsy contrivances.
An examination, in Figure 30, of the symbols for 1, 10, 100, and
1,000 indicates how the numerals are made up of straight lines. The
same is seen in 4, 40, 400, and 4,000 or in 5, 50, 500, and 5,000.
98. General remarks. — Evidently one of the earliest ways of re-
cording the small numbers, from 1 to 5, was by writing the corre-
sponding number of strokes or bars. To shorten the record in express-
ing larger numbers new devices were employed, such as placing the
bars representing higher values in a different position from the others,
or the introduction of an altogether new symbol, to be associated with
the primitive strokes on the additive, or multiplicative principle, or in
some cases also on the subtractive principle.
After the introduction of alphabets, and the observing of a fixed
sequence in listing the letters of the alphabets, the use of these letters
1 See M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. II (2d ed.;
Leipzig, 1913), p. 410; M. Cantor, Mathemat. Beitrdge zum Kulturleben der Volker
(Halle, 1863), p. 166, 167; G. Friedlein, Die Zahlzeichen und das elementare Rechnen
der Griechen und Romer (Erlangen, 1869), p. 12; T. H. Martin, Annali di mate-
maiica (B. Tortolini; Rome, 1863), Vol. V, p. 298; J. C. Heilbronner, Historia
Mathcseos universae (Lipsiae, 1742), p. 735-37; J. Ruska, Archivfiir die Geschichte
der Nalurwissenschaflen und Technik, Vol. IX (1922), p. 112-26.
HINDU-ARABIC NUMERALS
69
for the designation of numbers was introduced among the Syrians,
Greeks, Hebrews, and the early Arabs. The alphabetic numeral sys-
tems called for only very primitive powers of invention; they made
FIG. 30. — The numerals described by Noviomagus in 1539. (Taken from J. C.
Heilbronner, Historia malheseos [1742], p. 736.)
70 A HISTORY OF MATHEMATICAL NOTATIONS
unnecessarily heavy demands on the memory and embodied no at-
tempt to aid in the processes of computation.
The highest powers of invention were displayed in the systems em-
ploying the principle of local value. Instead of introducing new sym-
bols for units of higher order, this principle cleverly utilized the posi-
tion of one symbol relative to others, as the means of designating
different orders. Three important systems utilized this principle:
the Babylonian, the Maya, and the Hindu-Arabic systems. These
three were based upon different scales, namely, 60, 20 (except in one
step), and 10, respectively. The principle of local value applied to a
scale with a small base affords magnificent adaptation to processes of
computation. Comparing the processes of multiplication and division
which we carry out in the Hindu-Arabic scale with_what tEe alpha-
beticafsystems or the Roman system afforded places the superiority of
the Hindu-Arabic scale in full view. The Greeks resorted to abacal
computation, which is simply a primitive way of observing local value
in computation. In what way the Maya or the Babylonians used their
notations in computation is not evident from records that have come
down to us. The scales of 20 or 60 would crJl for large multiplication
tables.
The orjgjn_and development of the Hindu-Arabic notation has
received Intensive study. Nevertheless, little is known. An" outstand-
ing facTis~^Iit~"cIuffng "the past one thousand years no uniformity in
the shapes of the numerals has been reached. An American is some-
times puzzled by the shape of the number 5 written in France. A
European traveler in Turkey would find that what in Europe is a
0 is in Turkey a 5.
99. Opinion of Laplace. — Laplace1 expresses his admiration for the
invention of the Hindu-Arabic numerals and notation in this wise:
"It is from the Indians that there has come to us the ingenious method
of expressing all numbers, in ten characters, by giving them, at the
same time, an absolute and a place value; an idea fine and important,
which appears indeed so simple, that for this very reason we do not
sufficiently recognize its merit. But this very simplicity, and the
extreme facility which this method imparts to all calculation, place
jour system of arithmetic in the first rank of the useful inventions.
How difficult it was to invent such a method one can infer from the
fact that it escaped the genius of Archimedes and of Apollonius of
Perga, two of the greatest men of antiquity."
1 Exposition du systeme du monde (6th ed.; Paris, 1835), p. 376.
Ill
SYMBOLS IN ARITHMETIC AND ALGEBRA
(ELEMENTARY PART)
100. In ancient Babylonian and Egyptian documents occur cer-
tain ideograms and symbols which are not attributable to particular
individuals and are omitted here for that reason. Among these signs
is r~ for square root, occurring in a papyrus found at Kahun and now
at University College, London,1 and a pair of walking legs for squaring
in the Moscow papyrus.2 These symbols and ideograms will be referred
to in our "Topical Survey" of notations.
A. GROUPS OF SYMBOLS USED BY INDIVIDUAL WRITERS
GREEK: DIOPHANTUS, THIRD CENTURY A.D.
101. The unknown number in algebra, defined by Diophantus as
containing an undefined number of units, is represented by the Greek
letter s with an accent, thus s', or in the form s°'. In plural cases the
symbol was doubled by the Byzantines and later writers, with the
addition of case endings. Paul Tannery holds that the evidence is
against supposing that Diophantus himself duplicated the sign.3
G. H. F. Nesselmann4 takes this symbol to be final sigma and remarks
that probably its selection was prompted by the fact that it was the
only letter in the Greek alphabet which was not used in writing num-
bers. Heath favors "the assumption that the sign was a mere tachy-
graphic abbreviation and not an algebraical symbol like our x,
though discharging much the same function."6 Tannery suggests that
the sign is the ancient letter koppa, perhaps slightly modified. Other
views on this topic are recorded by Heath.
1 Moritz Cantor, Vorlesungen uber Geschichte der Malhematik, Vol. I, 3d ed.,
Leipzig, p. 94.
2 B. Touraeff, Ancient Egypt (1917), p. 102.
3 Diophanti Alcxandrini opera omnia cum Graedst commentaries (Lipsiae, 1895),
Vol. II, p. xxxiv-xlii; Sir Thomas L. Heath, Diophantus of Alexandria (2d ed.;
Cambridge, 1910),. p. 32, 33.
4 Die Algebra der Griechen (Berlin, 1842), p. 290, 291.
5 Op. cit.y p. 34-36.
71
72 A HISTORY OF MATHEMATICAL NOTATIONS
A square, z2, is in Diophantus' Arithmetica AF
A cube, x8, is in Diophantus' Arithmetica KY
A square-square, z4, is in Diophantus' Arithmetica ArA
A square-cube, z5, is in Diophantus' Arithmetica AKr
A cube-cube, x6, is in Diophantus' Arithmetica KrK
In place of the capital letters kappa and delta, small letters are some-
times used.1 Heath2 comments on these symbols as follows: "There is
no obvious connection between the symbol Ay and the symbol s
of which it is the square, as there is between x2 and x, and in this lies
the great inconvenience of the notation. But upon this notation no
advance was made even by late editors, such as Xylander, or by
Bachet and Fermat. They wrote N (which was short for Numerus) for
the s of Diophantus, Q (Quadratus) for A F, C (Cubus) for K y , so that we
find, for example, 1Q+5JV = 24, corresponding to z2+5z = 24.3 Other
symbols were however used even before the publication of Xylander's
Diophantus, e.g., in Bombelli's Algebra"
102. Diophantus has no symbol for multiplication; he writes down
the numerical results of multiplication without any preliminary step
which would necessitate the use of a symbol. Addition is expressed
1 From Format's edition of Bachct;s Diophantus (Toulouse, 1670), p. 2,
Definition II, we quote: "Appellatvr igitur Quadratus, Dynamis, & est illius nota
5' superscriptum habens u sic S«>. Qui autem sit ex quadrato in suum latus cubus
est, cuius nota est \, superscriptum habens v hoc pacto «w. Qui autem sit ex quad-
rato in seipsum multiplicato, quadrato-quadratus est, cuius nota est geminum 5'
habens superscriptum i>, hac ratione 55". Qui sit quadrato in cubum qui ab eodem
latere profectus est, ducto, quadrato-cubus nominatur, nota eius 5/c superscriptum
habens u sic 8i<y. Qui ex cubo in se ducto nascitur, cubocubus vocatur, & est eius
nota geminum K superscriptum habens v, hoc pacto KK". Cui vero nulla harum
proprietatum obtigit, sed constat multitudine vnitatem rationis experte, nurnerus
vocatur, nota eius V Est et aliud signum immutabile definitorum, vnitas, cuius
nota Jj. superscriptum habens 6 sic /z°." The passage in Bachet's edition of 1621 is
the same as this.
2 Op. tit., p. 38.
8 In Fermat's edition of Bachet's Diophantus (Toulouse, 1670), p. 3, Definition
II, we read: "Haec ad verbum exprimenda esse arbitratus sum potius quam cum
Xilandro nescio quid aliud comminisci. Quamuis enim in reliqua versione nostra
notis ab eodem Xilandro excogitatis libenter vsus sim, quas tradam infra. Hie
tamen ab ipso Diophanto longius recedere nolui, quod hac definitione notas ex-
plicet quibus passim libris istis vtitur ad species omnes compcndio designandas, &
qui has ignoret ne quidem Graeca Diophanti legere possit. Porr6 quadrat urn Dy-
namin vocat, quae vox potestatem sonat, quia videlicet quadratus est veluti
potestas cuius libet lineae, & passim ab Euclide, per id quod potest linea, quadratus
illius designatur. Itali, Hispanique eadem ferd de causa Censum vocant, quasi
INDIVIDUAL WRITERS 73
y mere juxtaposition. Thus the polynomial X3+13z2+5:r+2 would
o _ o
e in Diophantine symbols K FdA YLyseMp, where M is used to repre-
jnt units and shows that fi or 2 is the absolute term and not a part
f the coefficient of s or x. It is to be noted that in Diophantus'
square-cube" symbol for a;5, and "cube-cube" symbol for x6, the
Iditive principle for exponents is employed, rather than the multipli-
itive principle (found later widely prevalent among the Arabs and
^alians), according to which the "square-cube" power would mean x*
id the "cube-cube" would mean #9.
103. Diophantus' symbol for subtraction is "an inverted ^ with
le top shortened, A." Heath pertinently remarks: "As Diophantus
scd no distinct sign for +, it is clearly necessary, in order to avoid
mfusion, that all the negative terms in an expression, should be
laced together after all the positive terms. And so in fact he does
lace them."1 As regards the origin of this sign /jv, Heath believes
lat the explanation which is quoted above from the Diophantine
ixt as we have it is not due to Diophantus himself, but is "an explana-
on made by a scribe of a symbol which he did not understand."
eath2 advances the hypothesis that the symbol originated by placing
I within the uncial form A> thus yielding A . Paul Tannery,3 on the
;her hand, in 1895 thought that the sign in question was adapted
om the old letter sampi !), but in 1904 he4 concluded that it was
,ther a conventional abbreviation associated with the root of a cer-
in Greek verb. His considerations involve questions of Greek gram-
ar and were prompted by the appearance of the Diophantine sign
ms rcdditum, prouentumque, qudd a latere seu radice, tanquam a feraci solo
ladratus oriatur. Inde factum vt Gallorum nonnulli & Cermanorum corrupto
cabulo zerizum appellarint. Numerum autem indeterminatum & ignotum, qui
aliarum omnium potestatum latus esse intelligitur, Numerum sirnpliciter Dio-
antus appellat. Alij passim Radicem, vel latus, vel rein dixerunt, Itali patrio
cabulo Cosam. Caeterum nos in versione nostra his notis N. Q. C. QQ. QC. CC.
signabimus Numerum, Quadratum, Cubum, Quadratoquadratum, Quadrato-
bum, Cubocubum. Nam quod ad vnitates certas & determinatas spectat, eis
tarn aliquam adscribere superuacaneum duxi, qu6d hae seipsis absque vlla
ibiguitate sese satis indicent. Ecquis enim cum audit numerum 6. non statim
^itat sex vnitates? Quid ergo necesse est sex vnitates dicere, cum sufficiat dicere,
c? . . . . " This passage is the same as in Bachet's edition of 1621.
1 Heath, op. cit., p. 42.
2 Ibid., p. 42, 43.
3 Tannery, op. cit., Vol. II, p. xli.
4 Bibliolheca mathematica (3d ser.), Vol. V, p. 5-8.
74 A HISTORY OF MATHEMATICAL NOTATIONS
of subtraction in the critical notes to Schone's edition1 of the Metrica
of Heron.
For equality the sign in the archetypal manuscripts seems to have
been i°\ "but copyists introduced a sign which was sometimes con-
fused with the sign l|" (Heath).
104. The notation for division comes under the same head as the
notation for fractions (see § 41). In the case of unit fractions, a
double accent is used with the denominator: thus y" = %. Sometimes
a simple accent is used; sometimes it appears in a somewhat modified
form as ^, or (as Tannery interprets it) as X •' thus y^~ J . For \-
appear the symbols Z' and ^, the latter sometimes without the dot.
Of fractions that are not unit fractions, f has a peculiar sign U7 of its
own, as was the case in Egyptian notations. "Curiously enough,"
says Heath, "it occurs only four times in Diophantus." In some old
manuscripts the denominator is written above the numerator, in
some rare cases. Once we find ie8 = *45 , the denominator taking the
position where we place exponents. Another alternative is to write
the numerator first and the denominator after it in the same line,
marking the denominator with a submultiple sign in some form : thus,
=f y = | .2 The following are examples of fractions from Diophantus :
From v. 10: l^ = ~ From v. 8, Lemma: 0ZV = 2 -J- J
l £* \-£
8 V 250
From iv. 3: sX*/ = ~ From iv. 15: r^ —
Fromvi. 12: ^M
= (60z2+2,520)/(z4+900-60z2) .
105. The fact that Diophantus had only one symbol for unknown
quantity affected considerably his mode of exposition. Says Heath:
"This limitation has made his procedure often very different from our
modern work." As we have seen, Diophantus used but few symbols.
Sometimes he ignored even these by describing an operation in words,
when the symbol would have answered as well or better. Considering
the amount of symbolism used, Diophantus' algebra may be desig-
nated as "syncopated."
1 Heronis Alexandrini opera, Vol. Ill (Leipzig, 1903), p. 156, 1. 8, 10. The
manuscript reading is novkbuv oSriS', the meaning of which is 74 — jV
2 Heath, op. oil., p. 45, 47.
INDIVIDUAL WRITERS 75
HINDU: BRAHMAGUPTA, SEVENTH CENTURY A.D.
106. We begin with a quotation from H. T. Colebrooke on Hindu
algebraic notation:1 "The Hindu algebraists use abbreviations and
initials for symbols: they distinguish negative quantities by a dot,
but have not any mark, besides the absence of the negative sign, to
discriminate a positive quantity. No marks or symbols (other than
abbreviations of words) indicating operations of addition or multipli-
cation, etc., are employed by them: nor any announcing equality2
or relative magnitude (greater or less) A fraction is indicated
by placing the divisor under the dividend, but without a line of sepa-
ration. The two sides of an equation are ordered in the same manner,
one under the other The symbols of unknown quantity are not
confined to a single one: but extend to ever so great a variety of
denominations: and the characters used are the initial syllables of
the names of colours, excepting the first, which is the initial of ydvat-
tdvat, as much as."
107. In Brahmagupta,3 and later Hindu writers, abbreviations
occur which, when transliterated into our alphabet, are as follows:
ru for rupa, the absolute number
ya for ydvat-tdvat, the (first) unknown
ca for calaca (black), a second unknown
ni for nilaca (blue), a third unknown
pi for pitaca (yellow), a fourth unknown
pa for pandu (white), a fifth unknown
lo for lohita (red), a sixth unknown
c for caranij surd, or square root
ya v for x2, the v being the contraction for
varga, square number
108. In Brahmagupta,4 the division of ru 3 c 450 c 75 c 54 by
c 18 c 3 (i.e., 3+ V/450+l/ 75+1/54 by 1/18+1/3) is carried out as
follows: "Put c 18 c 3. The dividend and divisor, multiplied by this,
make ru 75 c 625. The dividend being then divided by the single surd
ru 15
constituting the divisor, the quotient is ru 5 c 3."
1 H. T. Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit
of Bramegupta and Bhdscara (London, 1817), p. x, xi.
2 The Bakhshali MS (§ 109) was found after the time of Colebrooke and has
an equality sign.
*Ibid., p. 339 ff.
4 Brahme-sphuta-sidd'hdnta, chap. xii. Translated by H. T. Colebrooke in
op. cit. (1817), p. 277-378; we quote from p. 342.
A HISTORY OF MATHEMATICAL NOTATIONS
| In modern symbols, the statement is, substantially: Multipl
Mend and divisor by 1/18— 1/3; the products are 75+1/675 an
15; divide the former by the latter, 5+1/3.
"Question 16.1 When does the residue of revolutions of the sur
less one, fall, on a Wednesday, equal to the square root of two leg
than the residue of revolutions, less one, multiplied by ten and aug
mented by two?
"The value of residue of revolutions is to be here put square c
ydvat-tdvat with two added : ya v 1 ru 2 is the residue of revolutions
Sanskrit character
or letters, by which the Hindus denote the unknown quan-
tities in their notation, are the following: TJJ,. offf,
FIG. 31. — Sanskrit symbols for unknowns. (From Charles Hutton, Mai In
matical Tracts, II, 167.) The first symbol, pa, is the contraction for "white"; th
second, ca, the initial for "black"; the third, ni, the initial for "blue"; the fourti
pi, the initial for "yellow"; the fifth, lo, for "red."
This less two isyavl; the square root of which isyal. Less one, it i
ya 1 ru 1; which multiplied by ten is ya 10 ru 10; and augmented fy
two, ya 10 ru 8. It is equal to the residue of revolutions yavl ru 2 less
1 a*. *. x f i xi • i ya v 0 ya 10 ru 8 ,,
one; viz. yav I ru 1. Statement of both sides ' ~ ., . li/qua
J ya v 1 ya 0 ru 1
subtraction being made conformably to rule 1 there arises ya v I
ya 10
Now, from the absolute number (9), multiplied by four times the [co
efficient of the] square (36), and added to (100) the square of the
[coefficient of the] middle term (making consequently 64), the square
root being extracted (8), and lessened by the [coefficient of the] middle
term (10), the remainder is 18 divided by twice the [coefficient of the
square (2), yields the value of the middle term 9. Substituting with
this in the expression put for the residue of revolutions, the answei
comes out, residue of revolutions of the sun 83. Elapsed period ol
days deduced from this, 393, must have the denominator in leasl
terms added so often until it fall on Wednesday."
1 Colebrooke, op. ciL, p. 346. The abbreviations ru, c, ya, ya v, ca, ni, etc., an
f rar»o1if r»ro f innu r\f fV«r* /irki'»«r>c'r\/~kr»/4ir»rr IriffoTO \r\ fl»o Sianalrrif alnVial-kftt
INDIVIDUAL WRITERS 77
Notice that ya V ® ya J? m J signifies Oz2+10z-8 = z2+0;r+l.
7/a v 1 ya 0 ru 1
Brahmagupta gives1 the following equation in three unknown
quantities and the expression of one unknown in terms of the other
two:
"ya 197 ca 1644 nil ru 0
ya 0 ca 0 ni 0 ru 6302.
Equal subtraction being made, the value of ydvat-tdvat is
ca 1644 ni 1 ru 6302 ."
M 197
In modern notation:
whence,
_1644?/+;g+6302
X~~ ~ 197
HINDU: THE BAKHSHALI MS
109. The so-called Bakhshali MS, found in 1881 buried in the
earth near the village of Bakhshali in the northwestern frontier of
India, is an arithmetic written on leaves of birch-bark, but has come
down in mutilated condition. It is an incomplete copy of an older
manuscript, the copy having been prepared, probably about the
eighth, ninth, or tenth century. "The system of notation/' says A. F.
Rudolph Hoernle,2 "is much the same as that employed in the arith-
metical works of Brahmagupta and Bhaskara. There is, however, a
very important exception. The sign for the negative quantity is a
cross (+). It looks exactly like our modern sign for the positive
quantity, but it is placed after the number which it qualifies. Thus
12 7 I
means 12—7 (i.e. 5). This is a sign which I have not met with
in any other Indian arithmetic ..... The sign now used is a dot placed
over the number to which it refers. Here, therefore, there appears to
be a mark of great antiquity. As to its origin I am not able to suggest
any satisfactory explanation ..... A whole number, when it occurs in
an arithmetical operation, as may be seen from the above given ex-
ample, is indicated by placing the number 1 under it. This, however, is
1 Colebrooke, op. cit., p. 352.
2 "The Bakhshali Manuscript," Indian Antiquary, Vol. XVII (Bombay, 1888),
p. 33-48, 275-79; see p. 34.
78
A HISTORY OF MATHEMATICAL NOTATIONS
a practice which is still occasionally observed in India The
following statement from the first example of the twenty-fifth siitra
affords a good example of the system of notation employed in the
Bakhshall arithmetic:
1
1 1 1 bhd 32
1 1 1
3+ 3+ 3+
phalarh 108
Here the initial dot is used much in the same way as we use the letter x
to denote the unknown quantity, the value of which is sought. The
number 1 under the dot is the sign of the whole (in this case, unknown)
number. A fraction is denoted by placing one number under the other
without any line of separation; thus Q is ,,, i.e. one-third. A mixed
o o
number is shown by placing the three numbers under one another;
1
,1
1
thus 1 is 1+,, or 10, i.e. one and one-third. Hence 1
3+
means 1 — ,3
o
i.e. - ). Multiplication is usually indicated by placing the numbers
side by side; thus
& 39.
phalam 20
5 32
8 1
1
o
means 0X32 = 20. Similarly 1
222
means .-;X.;X,-, or
333
8
3+ 3+ 3+
i.e. ~. Bhd is an abbreviation of bhdga, 'part/ and means that the
number preceding it is to be treated as a denominator. Hence
111 8 27
111 bhd means 1 4- ~= or -^-. The whole statement, therefore,
3+ 3+ 3+ Z7 *
1 1 1
1
1
1
1 bhd 32
3+ 3+ 3+
phalam 108 ,
27,
means Q X 32 = 108, and may be thus explained, — 'a certain number is
o
g
found by dividing with ^ and multiplying with 32; that number is
108.' The dot is also used for another purpose, namely as one of the
INDIVIDUAL WRITERS 79
ten fundamental figures of the decimal system of notation, or the
zero (0123456789). It is still so used in India for both purposes, to
indicate the unknown quantity as well as the naught The
Indian dot, unlike our modern zero, is not properly a numerical figure
at all. It is simply a sign to indicate an empty place or a hiatus. This
is clearly shown by its name sdnya, 'empty/ .... Thus the two fig-
ures 3 and 7, placed in juxtaposition (37), mean 'thirty-seven/ but
with an 'empty space' interposed between them (3 7), they mean
'three hundred and seven/ To
prevent misunderstanding the
presence of the 'empty space'
was indicated by a dot (3.7);
or by what is now the zero
(307). On the other hand, oc-
v ; . '
currmg in the statement of
a problem, the 'empty place7
could be filled up, and here the -n .J0 ^ „ , l l .,, .,, , .
^' FIG. 32. — From Bakhshah arithmetic
dot which marked its presence (G. R. Kay6j Indian Mathematics [1915],
signified a 'something' which p. 26; R. Hoernle, op. tit., p. 277).
was to be discovered and to
be put in the empty place In its double signification, which
still survives in India, we can still discern an indication of that
country as its birthplace The operation of multiplication
alone is not indicated by any special sign. Addition is indicated
by yu (for yuta), subtraction by + (ka for kanitaf) and division
by bhd (for bhdga). The whole operation is commonly enclosed be-
tween lines (or sometimes double lines), and the result is set down
outside, introduced by pha (for phald)." Thus, pha served as a sign
of equality.
The problem solved in Figure 32 appears from the extant parts
to have been : Of a certain quantity of goods, a merchant has to pay,
as duty, £, \, and J on three successive occasions. The total duty is
24. What was the original quantity of his goods? The solution ap-
pears in the manuscript as follows: "Having subtracted the series
from one," we get f , f , $ ; these multiplied together give | ; that again,
subtracted from 1 gives f; with this, after having divided (i.e., in-
verted, f), the total duty (24) is multiplied, giving 40; that is the
original amount. Proof: £ multiplied by 40 gives 16 as the remainder.
Hence the original amount is 40. Another proof: 40 multiplied by
1 — £ and 1 — } and 1 — J gives the result 16; the deduction is 24; hence
the total is 40.
80 A HISTORY OF MATHEMATICAL NOTATIONS
HINDU: BHASKARA, TWELFTH CENTURY A.D.
110. Bhaskara speaks in his Lilavati1 of squares and cubes of
numbers and makes an allusion to the raising of numbers to higher
powers than the cube. Ganesa, a sixteenth-century Indian commen-
tator of Bhaskara, specifics some of them. Taking the words varga for
square of a number, and g'hana for cube of a number (found in Bhas-
kara and earlier writers), Ganesa explains2 that the product of four
like numbers is the square of a square, varga-varga; the product of six
like numbers is the cube of a square, or square of a cube, varga-g'hana
or g'hana-varga; the product of eight numbers gives varga-varga-varga;
of nine, gives the cube of a cube, g'hana-g'hana. The fifth power was
called varga-g'hana-ghdta; the seventh, varga-varga-g'hana-ghdta.
111. It is of importance to note that the higher powers of the
unknown number are built up on the principle of involution, except
the powers whose index is a prime number. According to this prin-
ciple, indices are multiplied. Thus g'hana-varga does not mean n3-n2 =
n5, but (tt3)2 = n6. Similarly, g'hana-g'hana does not mean n3«n3=n6,
but (n3)3 = n9. In the case of indices that are prime, as in the fifth and
seventh powers, the multiplicative principle became inoperative and
the additive principle was resorted to. This is indicated by the word
ghdta ("product")- Thus, varga-g'hana-ghdta means n2'n3 = n5.
In the application, whenever possible, of the multiplicative prin-
ciple in building up a symbolism for the higher powers of a number, we
see a departure from Diophantus. With Diophantus the symbol for
x2, followed by the symbol for x3, meant x5; with the Hindus it meant
x6. We shall see that among the Arabs and the Europeans of the
thirteenth to the seventeenth centuries, the practice was divided,
some following the Hindu plan, others the plan of Diophantus.
112. In Bhaskara, when unlike colors (dissimilar unknown quanti-
ties, like x and y) are multiplied together, the result is called bhavita
("product"), and is abbreviated bha. Says Colebrooke: "The prod-
uct of two unknown quantities is denoted by three letters or syllables,
as ya.ca bha, ca.ni bha, etc. Or, if one of the quantities be a higher
power, more syllables or letters are requisite; for the square, cube,
etc., are likewise denoted by the initial syllables, va, gha, va-va, va-gha,
gha-gha,* etc. Thus ya va • ca gha bha will signify the square of the
1 Colebrooke, op. cit., p. 9, 10.
*Ibid., p. 10, n.3;p. 11.
8 Gha-gha for the sixth, instead of the ninth, power, indicates the use here of the
additive principle.
INDIVIDUAL WRITERS
81
first unknown quantity multiplied by the cube of the second. A dot
is, in some copies of the text and its commentaries, interposed between
the factors, without any special direction, however, for this notation."1
Instead of ya va one finds in Brahmagupta and BhiLskara also the
severer contraction ya v; similarly, one finds cav for the square of the
second unknown.2
It should be noted also that "equations are not ordered so as to
put all the quantities positive; nor to give precedence to a positive
term in a compound quantity: for the negative terms are retained,
and even preferably put in the first place."3
According to N. Raman ujacharia and G. R. Kaye,4 the content of
the part of the manuscript shown in Figure 33 is as follows: The
*rtrv
,-*
jSg^g^aS
^^»^**^
S&iEHE^')^aft
[b^Uf\lifr&%Jd&uJU^I«fc
©i^tih^^^g^t
8feUtfi%it / ' ._
^*^kk^^^?i^^''^/6^>g
FIG. 33. — Sridhara's Trisdtika. Sridhara was born 991 A.D. Ho is cited by
Bhaskara; he explains the "Hindu method of completing the square" in solving
quadratic equations.
circumference of a circle is equal to the square root of ten times the
square of its diameter. The area is the square root of the product of
ten with the square of half the diameter. Multiply the quantity whose
square root cannot be found by any large number, take the square
root of the product, leaving out of account the remainder. Divide
it by the square root of the factor. To find the segment of a circle,
take the sum of the chord and arrow, multiply it by the arrow, and
square the product. Again multiply it by ten-ninths and extract its
square root. Plane figures other than these areas should be calculated
by considering them to be composed of quadrilaterals, segments of
circles, etc.
1Op. cit., p. 140, n. 2; p. 141. In this quotation we omitted, for simplicity,
some of the accents found in Colebrooke's transliteration from the Sanskrit.
2 Ibid., p. 63, 140, 346.
3 Ibid., p. xii.
4 Bibliotheca mathematica (3d ser.), Vol. XIIT (1912-13), p. 206, 213, 214.
82 A HISTORY OF MATHEMATICAL NOTATIONS
113. Bhaskara Achdbrya, "Lilavati,"1 11 50 A D— "Example: Tell
me the fractions reduced to a common denominator which answer to
three and a fifth, and one-third, proposed for addition; and those
which correspond to a sixty-third and a fourteenth offered for sub-
traction. Statement:
3 1 1
1 5 3
Answer: Reduced to a common denominator
45 3 5 G 53
15 15 15' bumi5-
Statement of the second example:
1 1
63 14 '
Answer: The denominator being abridged, or reduced to least terms,
by the common measure seven, the fractions become
1 1
9 2*
Numerator and denominator, multiplied by the abridged denomina-
p
2 9
tors, give respectively . ^p anc^ i or • Subtraction being made, the
difference is .™ .
114. Bhaskara Achdbrya, "Vija-Ganita."2— "Example: Tell
quickly the result of the numbers three and four, negative or affirma-
tive, taken together: .... The characters, denoting the quantities
known and unknown, should be first written to indicate them gener-
ally; and those, which become negative, should be then marked with
a dot over them. Statement:3 3*4. Adding them, the sum is found 7.
Statement: 3»4. Adding them, the sum is 7. Statement: 3*4. Tak-
ing the difference, the result of addition comes out 1.
" 'So much as' and the colours 'black, blue, yellow and red/4 and
others besides these, have been selected by venerable teaShers for
names of values of unknown quantities, for the purpose of reckoning
therewith.
1 Colebrooke, op. cit., p. 13, 14. 2 Ibid., p. 131.
3 In modern notation, 3+4 = 7, (-3) + (-4) = -7, 3 + (-4) = -1.
4 Colebrooke, op. cit., p. 139.
INDIVIDUAL WRITERS 83
"Example:1 Say quickly, friend, what will affirmative one un-
known with one absolute, and affirmative pair unknown less eight
absolute, make, if addition of the two sets take place? .... State-
ment :2
ya 1 ru 1
ya 2 ru 8
Answer: The sum is ya 3 ru 7.
"When absolute number and colour (or letter) are multiplied one
by the other, the product will be colour (or letter). When two, three
or more homogeneous quantities are multiplied together, the product
will be the square, cube or other [power] of the quantity. But, if
unlike quantities be multiplied, the result is their (bhdvita) 'to be'
product or factum.
"23. Example:3 Tell directly, learned sir, $he product of the
multiplication of the unknown (ydvat-tdvat) five, less the absolute num-
ber one, by the unknown (ydvat-tdvat) thrfce joined with the absolute
two: .... Statement:4
ya 5 ru 1 ^ , . . _ _ •
0 0 Product: ya v 15 ya 1 ru 2 .
ya 3 ru 2 J u
"Example:5 'So much as' three, 'black' five, 'blue' seven, all
affirmative: how many do they make with negative two, three, and
one of the same respectively, added to or subtracted from them?
Statement:6
ya 3 ca 5 ni 7 Answer: Sum ya I ca 2 ni 6 .
ya 2 ca 3 ni 1 Difference ya 5 ca 8 ni 8 .
"Example:7 Say, friend, [find] the sum and difference of two ir-
rational numbers eight and two: .... after full consideration, if thou
be acquainted with the sixfold rule of surds. Statement :8 c 2 c 8.
1 Ibid. 2 In modern notation, x-\- 1 and 2x— 8 have the sum 3x— 7.
3 Colebrookc, op. cit., p. 141, 142.
4 In modern notation (5x - 1) (3x +2) = 15x2 +7x -2.
5 Colcbrooke, op. cit., p. 144.
6 In modern symbols, 3x -\-5y-\-7z and — 2x— 3y—z have the sum z-f 2y+Gz,
and the difference 5z-f 8?/-f 8z.
7 Colebrooke, op. cit., p. 146.
8 In modern symbols, the example is l/8+/2 = vT8, T/8-1/^/2. The
same example is given earlier by Brahmagupta in his Brahme-sputOrsidd'hdnta,
chap, xviii, in Colebrooke, op. cit., p. 341.
84 A HISTORY OF MATHEMATICAL NOTATIONS
Answer: Addition being made, the sum is c 18. Subtraction taking
place, the difference is c 2."
ARABIC: aL-KHOWARizMi, NINTH CENTURY A.D.
115. In 772 Indian astronomy became known to Arabic scholars.
As regards algebra, the early Arabs failed to adopt either the Dio-
phantine or the Hindu notations. The famous Algebra of al-Khow£r-
izmi of Bagdad was published in the original Arabic, together with an
English translation, by Frederic Rosen,1 in 1831. He used a manu-
script preserved in the Bodleian Collection at Oxford. An examination
of this text shows that the exposition was altogether rhetorical, i.e.,
devoid of all symbolism. "Numerals arc in the text of the work al-
ways expressed by words: [Hindu-Arabic] figures are only used in
some of the diagrams, arid in a few marginal notes."2 As a specimen
of al-Khowarizmi's exposition we quote the following from his Algebra,
as translated by Rosen:
"What must be the amount of a square, which, when twenty-one
dirhems are added to it, becomes equal to the equivalent of ten roots
of that square? Solution: Halve the number of the roots; the moiety
is five. Multiply this by itself; the product is twenty-five. Subtract
from this the twenty-one which are connected with the square; the
remainder is four. Extract its root; it is two. Subtract this from the
moiety of the roots, which is five; the remainder is three. This is the
root of the square which you required, and the square is nine. Or you
may add the root to the moiety of the roots; the sum is seven; this is
the root of the square which you sought for, and the square itself is
forty-nine."3
By way of explanation, Rosen indicates the steps in this solution,
expressed in modern symbols, as follows: Example:
ARABIC: aL-KARKHi, EARLY ELEVENTH CENTURY A.D.
116. It is worthy of note that while Arabic algebraists usually
build up the higher powers of the unknown quantit}' on the multiplica-
tive principle of the Hindus, there is at least one Arabic writer, al-
Karkhi of Bagdad, who followed the Diophantine additive principle.4
1 The Algebra of Mohammed Ben Musa (cd. and trans. Frederic Rosen; London,
1831). See also L. C. Karpinski, Robert of Chester's Latin Translation of the Algebra
of Al-Khowarizmi (1915).
2 Rosen, op. dt.., p. xv. 3 Ibid., p. 11.
4 See Cantor, op. cit., Vol. I (3d ed.), p. 767, 768; Heath, op. dt.t p. 41.
INDIVIDUAL WRITERS 85
In al-Kharki's work, the Fakhri, the word mal means x2, kacb means
a3; the higher powers are mal mal for x4, mdl kacb for x5 (not for x6),
kaLb kacb for x6 (not for x9), wdZ moZ kacb for x7 (not for x12), and so on.
Cantor1 points out that there are cases among Arabic writers
where mdl is made to stand for x, instead of x2, and that this ambiguity
is reflected in the early Latin translations from the Arabic, where the
word census sometimes means x, and not x2.2
BYZANTINE: MICHAEL PSELLUS, ELEVENTH CENTURY A.D.
117. Michael Psellus, a Byzantine writer of the eleventh century
who among his contemporaries enjoyed the reputation of being the
first of philosophers, wrote a letter3 about Diophantus, in which he
gives the names of the successive powers of the unknown, used in
Egypt, which are of historical interest in connection with the names
used some centuries later by Nicolas Chuquet and Luca Pacioli. In
Psellus the successive powers are designated as the first number, the
second n umber (square), etc. This nomenclature appears to have been
borrowed, through the medium of the commentary by Hypatia, from
Anatolius, a contemporary of Diophantus.4 The association of the
successive powers of the unknown with the series of natural numbers
is perhaps a partial recognition of exponential values, for which there
existed then, and for several centuries that followed Psellus, no ade-
quate notation. The next power after the fourth, namely, x5, the
Egyptians called "the first undescribed," because it is neither a
square nor a cube; the sixth power they called the "cube-cube"; but
the seventh was "the second undescribed," as being the product of
the square and the "first undescribed." These expressions for x6 and
x7 are closely related to Luca Pacioli's primo relato and secondo relato,
found in his Summa of 1494.5 Was Pacioli directly or indirectly in-
fluenced by Michael Psellus?
ARABIC: IBN ALBANNA, THIRTEENTH CENTURY A.D.
118. While the early Arabic algebras of the Orient are character-
ized by almost complete absence of signs, certain later Arabic works on
1 Op. tit., p. 768. See also Karpinski, op. tit., p. 107, n. 1.
2 Such translations are printed by G. Libri, in his Histoire des stiences matht-
matiques, Vol. I (Paris, 1838), p. 276, 277, 305.
3 Reproduced by Paul Tannery, op. tit., Vol. II (1895), p. 37-42.
4 See Heath, op. tit., p. 2, 18.
5 See ibid., p. 41; Cantor, op. tit., Vol. II (2d ed.), p. 317.
86 A HISTORY OF MATHEMATICAL NOTATIONS
algebra, produced in the Occident, particularly that of al-Qalasadi of
Granacla, exhibit considerable symbolism. In fact, as early as the
thirteenth century symbolism began to appear; for example, a nota-
tion for continued fractions in al-Ha$sar (§391). Ibn Khaldun1
states that Ibn Albanna at the close of the thirteenth century wrote a
book when under the influence of the works of two predecessors, Ibn
Almuncim and Alahdab. "He [Ibn Albanna] gave a summary of the
demonstrations of these two works and of other things as well, con-
cerning the technical employment of symbols2 in the proofs, which
serve at the same time in the abstract reasoning and the representa-
tion to the eye, wherein lies the secret and essence of the explication
of theorems of calculation with the aid of signs." This statement of
Ibn Khaldun, from which it would seem that symbols were used by
Arabic mathematicians before the thirteenth century, finds apparent
confirmation in the translation of an Arabic text into Latin, effected
by Gerard of Cremona (1114-87). This translation contains symbols
for x and x2 which we shall notice more fully later. It is, of course,
quite possible that these notations were introduced into the text by
the translator and did not occur in the original Arabic. As regards
Ibn Albanna, many of his writings have been lost and none of his
extant works contain algebraic symbolism.
CHINESE: cnu SHIH-CHIEH
(1303 A.D.)
119. Chu Shih-Chieh bears the distinction of having been "in-
strumental in the advancement of the Chinese abacus algebra to the
highest mark it has ever attained."3 The Chinese notation is interest-
ing as being decidedly unique. Chu Shih-Chieh published in 1303 a
treatise, entitled Szu-yuen Yii-chien, or "The Precious Mirror of the
Four Elements," from which our examples are taken. An expression
like a+6+c+d, and its square, a2+62+c2+d2+2a6+2ac+2ad+
1 Consult F. Woepcke, "Rechcrches sur Fhistoire dcs sciences mathematiques
chez les orientaux," Journal asiatique (5th ser.), Vol. IV (Paris, 1854), p. 369-72;
Woepckc quotes the original Arabic and gives a translation in French. See also
Cantor, op, cit., Vol. I (3d ed,), p. 805.
2 Or, perhaps, letters of the alphabet.
3 Yoshio Mikami, The Development of Mathematics in China and Japan (Leip-
zig, 1912), p. 89. All our information relating to Chinese algebra is drawn from
this book, p. 89-98.
INDIVIDUAL WRITERS
87
2bc+2bd+2cd, were represented as shown in the following two illus-
trations:
1
1 202
2
1*1 1 0 -X- 0 1
2
1 202
1
Where we have used the asterisk in the middle, the original has the
character t'ai ("great extreme"). We may interpret this symbolism
by considering a located one space to the right of the asterisk (•#),&
above, c to the left, and d below. In the symbolism for the square of
a+b+c+d, the O's indicate that the terms a, 6, c, d do not occur in
the expression. The squares of these letters are designated by the 1's
two spaces from -K-. The four 2's farthest from -)£ stand for 2ab, 2ac,
2bc, 2bd, respectively, while the two 2's nearest to -)f stand for 2ac and
2bd. One is impressed both by the beautiful symmetry and by the
extreme limitations of this notation.
120. Previous to Chu Shih-Chieh's time algebraic equations of
only one unknown number were considered ; Chu extended the process
to as many as four unknowns. These unknowns or elements were
called the "elements of heaven, earth, man, and thing." Mikami
states that, of these, the heaven clement was arranged below the
known quantity (which was called "the great extreme"), the earth
clement to the left, the man element to the right, and the thing ele-
ment above. Letting -)f stand for the great extreme, and x, y, z, u, for
heaven, earth, man, thing, respectively, the idea is made plain by the
following representations :
Mikami gives additional illustrations:
0
0
1
1
0-2
#
0
1
0
1
+2yz
xz+z2
88
A HISTORY OF MATHEMATICAL NOTATIONS
Using the Hindu-Arabic numerals in place of the Chinese calculating
pieces or rods, Mikami represents three equations, used by Chu, in the
following manner:
In our notation, the four equations are, respectively,
a) 2x —
6) z2+ i/2- 22 = 0,
c) 2x +2y - u = 0.
No sign of equality is used here. All terms appear on one side of the
equation. Notwithstanding the two-dimensional character of the
notation, which permits symbols to be placed above and below the
starting-point, as well as to left and right, it made insufficient pro-
vision for the representation of complicated expressions and for easy
methods of computation. The scheme does not lend itself easily to
varying algebraic forms. It is difficult to see how, in such a system,
the science of algebra could experience a rapid and extended growth.
The fact that Chinese algebra reached a standstill after the thirteenth
century may be largely due to its inelastic and faulty notation.
BYZANTINE: MAXIMUS PLANUDES, FOURTEENTH CENTURY A.D.
121. Maximus Planudes, a monk of the first half of the fourteenth
century residing in Constantinople, brought out among his various
compilations in Greek an arithmetic,1 and also scholia to the first two
books of Diophantus' Arithmetical These scholia are of interest to us,
for, while Diophantus evidently wrote his equations in the running text
and did not assign each equation a separate line, we find in Planudes
the algebraic work broken up so that each step or each equation
is assigned a separate line, in a manner closely resembling modern
practice. To illustrate this, take the problem in Diophantus (i. 29),
1 Das Recheribuch des Maximus Planudes (Halle: herausgegeben von C. I.
Gcrhardt, 1865).
2 First printed in Xylander's Latin translation of Diophantus' Arilhmetica
(Basel, 1575). These scholia in Diophantus are again reprinted in P. Tannery,
Diophanti Alexandrini opera omnia (Lipsiae, 1895), Vol. II, p. 123-255; the ex-
ample which we quote is from p. 201.
INDIVIDUAL WRITERS 89
"to find two numbers such that their sum and the difference of their
squares are given numbers." We give the exposition of Planudes and
its translation.
Planudes Translation
K TT [Given the numbers], 20, 80
eK& • sd/x°Z M0*Asa Putting for the numbers, x+10,
10-z
rerp • AFdss/c^°p A Yd{jL°p A ss/c . . .Squaring, £2+20#+100,
z2+100-20z
virepox* ssju lff" JU°TT Taking the difference, 40# = 80
/xep • sd I9* jjpp Dividing, x = 2
for • M°^ AM?' Result, 12, 8
ITALIAN: LEONARDO OF PISA
(1202 A.D.)
122. Leonardo of Pisa's mathematical writings are almost wholly
rhetorical in mode of exposition. In his Liber abbaci (1202) he used the
Hindu-Arabic numerals. To a modern reader it looks odd to see
expressions like -£$ "A t 42, the fractions written before the integer in
the case of a mixed number. Yet that mode of writing is his invariable
practice. Similarly, the coefficient of x is written after the name for x,
as, for example,1 — "radices ^12" for I2%x. A computation is indi-
cated, or partly carried out, on the margin of the page, and is inclosed
in a rectangle, or some irregular polygon whose angles are right angles.
The reason for the inverted order of writing coefficients or of mixed
numbers is due, doubtless, to the habit formed from the study of
Arabic works; the Arabic script proceeds from right to left. Influ-
enced again by Arabic authors, Leonardo uses frequent geometric
figures, consisting of lines, triangles, and rectangles to illustrate
his arithmetic or algebraic operations. He showed a partiality for unit
fractions; he separated the numerator of a fraction from its denomi-
nator by a fractional line, but was probably not the first to do this
(§ 235). The product of a and b is indicated by factus ex.a.b. It has
boon stated that he denoted multiplication by juxtaposition,2 but
G. Enestrom shows by numerous quotations from the Liber abbaci
that such is not the case.3 Cantor's quotation from the Liber abbaci,
1 II liber abbaci di Leonardo Pisano (ed. B. Boncornpagni), Vol. I (Rome, 1867),
p. 407.
2 Cantor, op. tit., Vol. II (2d ed.), p. 62.
3 BMiotheca mathematica (3d ser.), Vol. XII (1910-11), p. 335, 336.
90 A HISTORY OF MATHEMATICAL NOTATIONS
"sit numerus .a.e.c. quaedam coniunctio quae uocetur prima, numeri
vero .d.b.f. sit coniunctio secunda,"1 is interpreted by him as a product,
the word coniunctio being taken to mean "product." On the other
hand, Kncstrom conjectures that numerus should be numeri, arid trans-
lates the passage as meaning, "Let the numbers a, e, c be the first, the
numbers d, 6, / the second combination." If Enestrom's interpreta-
tion is correct, then a.e.c and d.b.f are not products. Leonardo used in
his Liber abbaci the word res for x, as well as the word radix. Thus,
he speaks, "et intellige pro re summam aliquarn ignotam, quarn
inuenire uis."2 The following passage from the Liber abbaci contains
the words numerus (for a given number), radix for x, and census for x2:
"Primus cniin modus est, quando census et radices equantur numero.
.... Verbi gratia: duo census, et decem radices equantur denariis
30,"3 i.e., 2x'2+10x = 30. The use of res for x is found also in a Latin
translation of al-Khowarizmi's algebra,4 due perhaps to Gerard of
Cremona, where we find, "res in rein fit census/' i.e., x.x = x2. The
word radix for x as well as res, and substantia for a;2, are found in
Robert of Chester's Latin translation of al-Khowarizmi's algebra.5
Leonardo of Pisa calls x3 cubus, x* census census, XQ cubus cubus, or else
census census census; he says, " . . . . est multiplicare per cubum cubi,
sicut multiplicare per censum census census."6 He goes even farther
and lets x8 be census census census census. Observe that this phrase-
ology is based on the additive principle x2 • y? • x2 • x2 = xs. Leonardo
speaks also of radix census census.7
The first appearance of the abbreviation R or H for radix is in his
Pradica geometriae (1220),8 where one finds the R meaning "square
root" in an expression "et minus I}. 78125 dragme, et diminuta radice
28125 dragme." A few years later, in Leonardo's Flos,9 one finds
marginal notes which are abbreviations of passages in the text relating
to square root, as follows:
1 Op. cit., Vol. I (3d ed.), p. 132.
2 Ibid., Vol. I, p. 191.
3 Ibid., Vol. I, p. 407.
4 Libri, Histmre des sciences mathematiques en Italic, Vol. I (Paris, 1838), p. 268.
6 L. C. Karpinski, op. cit., p. 68, 82.
6 II liber abbaci, Vol. I, p. 447.
7 Ibid., Vol. I, p. 448.
8 Scritti di Leonardo Pisano (ed. B. Boncompagni), Vol. II (Rome, 1862), p.
209.
9 Op. cit., Vol. II, p. 231. For further particulars of the notations of Leonardo
of Pisa, see our §§ 219, 220, 235, 271-73, 290, 292, 318, Vol. II, §389. •
INDIVIDUAL WRITERS 91
.R.x p{. Bino.ij for primi [quidem] binomij radix
2 . ' B . R . x for radix [quippe] secundi binomij
.Bi. 3{. R.x for Tertij [autem] binomij radix
.Bi. 4l. R.x for Quarti [quoque] binomij radix
FRENCH: NICOLE ORESME, FOURTEENTH CENTURY A.D.
123. Nicole Oresme (ca. 1323-82), a bishop in Normandy, pre-
pared a manuscript entitled Algorismus proportionum, of which several
copies arc extant.1 He was the first to conceive the notion of fractional
powers which was afterward rediscovered by Stevin. More than this,
he suggested a notation for fractional powers. He considers powers of
ratios (called by him proportiones) . Representing, as does Oresme
himself, the ratio 2:1 by 2, Oresme expresses 2* by the symbolism
1 v
- ~ and reads this medietas [proportionis] duplae; he expresses
(20* by the symbolism
l.p.l
4.2.2
and reads it quarta pars [proportionis]
duplae sesquialterae. The fractional exponents { and | are placed to the
left of the ratios affected.
H. Wicleitner adds that Oresme did not use these symbols in com-
putation. Thus, Oresme expresses in words, ". . . . proponatur pro-
portio, que sit due tertie quadruple; et quia duo est numerator, ipsa
erit vna tertia quadruple duplicate, sev sedecuple,"2 i.e., 4§ = (42)* = 16*.
Oresme writes3 also: "Sequitur quod .a. moueatur velocius .6. in pro-
portione, que est medietas proportionis .50. ad .49.," which means,
"the velocity of a -.velocity of 6 = 1/50:1/49," the word medietas mean-
ing "square root."4
The transcription of the passage shown in Figure 34 is as follows:
''Una media debet sic scribi
una tertia sic
et due tertie
sic
et sic de alijs. et numerus, qui supra uirgularn, dicitur
1 Maximilian Curtze brought out an edition after the MS R. 4° 2 of the Gyrn-
nasiat-Bihliothck at Thorn, under the title Der Algorithmic Proportionum des
Nicolaus Oresme (Berlin, 1868). Our photographic illustration is taken from that
publication.
2 Curtze, op. cit., p. 15. 3 Ibid., p. 24.
4 See Knestrom, op. tit., Vol. XII (1911-12), p. 181. For further details see
also Curtze, Zeitschrift fur Mathematik und Phijsik, Vol. XIII (Suppl. 18G8),
p. 65 flf.
92
A HISTORY OF MATHEMATICAL NOTATIONS
numerator, iste uero, qui est sub uirgula, dicitur denominator. 2. Pro-
portio dupla scribitur isto modo 2.Za, et tripla isto modo 3.Za; et sic
de alijs. Proportio sesquialtera sic scribitur
, et sesquitertia
1 3
. Proportio superpartiens duas tertias scribitur sic
P2
13.
Proportio dupla superpartiens duas quartas scribitur sic
P2
24
;et
sic de alijs. 3. Medietas duple scribitur sic
Ip
2-2
, quarta pars
duple sesquialtere scribitur sic
4-2-2
; et sic de alijs."
FIG. 34. — From the first page of Oresme's Algorismus proportionum (four-
teenth century).
A free translation is as follows:
"Let a half be written
, a third ~ , and two-thirds
and so on. And the number above the line is called the 'numerator/
the one below the line is called the 'denominator.1 2. A double ratio
is written in this manner 2.Zo, a triple in this manner 3.'°, and thus in
other cases. The ratio one and one-half is written
, and one and
one-third is written
. The ratio one and two-thirds is written
. A double ratio and two-fourths are written
, and thus
INDIVIDUAL WRITERS 93
in other cases. 3. The square root of two is written thus
l.p
2 2
, the
thus
, and
fnnrfli rnof" rvf fwn pnH nnp-Vnlf is wriffpn fHii«s 1 - *
1U HI I'll ivJvfti \JL \j\\\J clillvl \JllC/HCvll ID Wllvvdl 1/llLlo I . ^ -.
1 4. 2. 2i
in other cases."
ARABIC: AL-QALASAD!, FIFTEENTH CENTURY A.D.
124. Al-Qalasadi's Raising of the Veil of the Science of Gubar ap-
peared too late to influence the progress of mathematics on the
European Continent. Al-Qalasadf used -^, the initial letter in the
Arabic word jidr, "square root"; the symbol was written above the
number whose square root was required and was usually separated
from it by a horizontal line. The same symbol, probably considered
this time as the first letter mjahala ("unknown"), was used to repre-
sent the unknown term in a proportion, the terms being separated by
the sign .'. . But in the part of al-Qalasadirs book dealing more par-
ticularly with algebra, the unknown quantity x is represented by the
letter \Jp, x2 by the letter \jo, x3 by the letter $"; these are written
above their respective coefficients. Addition is indicated by juxta-
position. Subtraction is *tfj ; the equality sign, J , is seen to resem-
ble the Diophantine t, if we bear in mind that the Arabs wrote from
right to left, so that the curved stroke faced in both cases the second
member of the equation. We reproduce from Woepcke's article a few
samples of al-Qalasadi's notation. Observe the peculiar shapes of the
Hindu- Arabic numerals (Fig. 35).
Woepcke1 reproduces also symbols from an anonymous Arabic
manuscript of unknown date which uses symbols for the powers of x
and for the powers of the reciprocal of x, built up on the additive prin-
ciple of Diophantus. The total absence of data relating to this manu-
script diminishes its historic value.
GERMAN: REGIOMONTANUS
(ca. 1473)
125. Regiomontanus died, in the prime of life, in 1476. After
having studied in Rome, he prepared an edition of Ptolemy2 which
was issued in 1543 as a posthumous publication. It is almost purely
rhetorical, as appears from the following quotation on pages 21 and 22.
1 Op. cit., p. 375-80.
2 loannis de Monte Regio et Georgii Pvrbachii epitome, in Cl. Ptolemaei magnam
compositionem (Basel, 1543). The copy examined belongs to Mr. F. E. Brasch.
94 A HISTORY OF MATHEMATICAL NOTATIONS
By the aid of a quadrant is determined the angular elevation ACE,
"que erit altitude tropici hiemalis," and the angular elevation ACFy
"que erit altitudo tropici aestivalis," it being required to find the arc
EF between the two. "Arcus itaque EFy fiet distantia duorum tropi-
9
&
1
—
... ^; \/54... ^; j
FORMULES D^QUATIONS TRINOMES.
2.
(ft,
PROPORTIONS.
7 : 12 = 84 : x ..... ^ .
ii ; 20 == 66 : « ...... a^. A 6:6 .-. &>o A || .
Fir,. 35. — Al-Qalasddi's algebraic symbols. (Compiled by F. Woepcko, Journal
asialique [Oct. and Nov., 1854], p. 363, 364, 366.)
corum quesita. Hac Ptolemaeus reperit 47. graduum 42. minutorum
40. secundorum. Inuenit enim proportionem eius ad totum circulu sicut
11. ad 83, postea uero minorem inuenerunt. Nos autem inuenimus
arcum AF 65. graduum 6. minutorum, & arcum AE 18. graduum 10.
INDIVIDUAL WRITERS 95
minutorum. Ideoq. nunc distantia tropicorum est 46. graduum 56.
niinutorum, ergo declinatio soils maxima nostro tempore est 23.
graduum 28. minutorum. "
126. We know, however, that in some of his letters and manu-
scripts symbols appear. They are found in letters and sheets contain-
ing computations, written by Regiomontanus to Giovanni Bianchini,
Jacob von Speier, and Christian Roder, in the period 1463-71. These
documents are kept in the Stadtbibliothek of the city of Niirnberg.1
Regiomontanus and Bianchini designate angles thus: gr 35 m 17;
Regiomontanus writes also: 44. 42'. 4" (see also § 127).
In one place2 Regiomontanus solves the problem: Divide 100 by a
certain number, then divide 100 by that number increased by 8;
the sum of the quotients is 40. Find the first divisor. Regiomontanus
writes the solution thus :
In Modern Symbols
"100 _JP9_ 100 JLOO
\TJe l^et'8 x x+8
100^ et 800 100z+800
800 200S+800
sf+Sx
40c£ et 320^ — 200^ et 800 40x2+320x = 200^+800.
40c£ et 120 if — 800 40x2+ I20x = 800
IcCet 3^f — 20 x2+ 3z = 20
I • J- addo numerum 20!J— 8/- f 1 !} add the no. 20J- = *£•
Radix quadrata de -849 minus | — 1 ^ l^8/ — * = x
Primus ergo divisor fuit 1} de 22 j Hence the first divisor was
I? 11." 1/221 -H.
Note that "plus" is indicated here by et; "minus" by 19, which is
probably a ligature or abbreviation of "minus." The unknown quan-
tity is represented by ^ and its square by c£- Besides, he had a sign
for equality, namely, a horizontal dash, such as was used later in
Italy by Luca Pacioli, Ghaligai, and others. See also Fig. 36.
1 Curtze, Urkunden zur Gcschichte der Mathemalik im Mittelalter und der Re-
naissance (Leipzig, 1902), p. 185-336 = A bhandlungen zur Geschichle der Mathe-
matik, Vol. XII. Sec -also L. C. Karpinski, Robert of Chester's Translation of the
Algebra of Al-Khowarizmi (1915), p. 36, 37.
2 Curtze, op. tit., p. 278.
96 A HISTORY OF MATHEMATICAL NOTATIONS
127. Figure 371 illustrates part of the first page of a calendar issued
by Regiomontanus. It has the heading Janer (" January "). Farther
to the right are the words Sunne — Monde — Stainpock ("Sun — Moon —
Capricorn")- The first line is 1 A. Kl. New Jar (i.e., "first day, A.
calendar, New Year"). The second line is 2. b. 4. no. der achtet S.
Stephans. The seven letters A , 6, c, dy e, F, g, in the second column on the
left, are the dominical letters of the calendars. Then come the days
of the Roman calendar. After the column of saints' days comes a
double column for the place of the sun. Then follow two double
columns for the moon's longitude; one for the mean, the other for the
10 -x
100- 10*
s*- Wx
% x* + 101) - 20x
FIG. 36. — Computations of Regiomontanus, in letters of about 1460. (From
manuscript, Niirnberg, fol. 23. (Taken from J. Tropfke, Geschichte der Elementar-
Mathematik (2d ed.), Vol. II [1921], p. 14.)
true. The S signifies signum (i.e., 30°); the G signifies gradus, or
"degree." The numerals, says De Morgan, are those facsimiles of the
numerals used in manuscripts which are totally abandoned before
the end of the fifteenth century, except perhaps in reprints. Note
the shapes of the 5 and 7. This almanac of Regiomontanus and the
Compotus of Anianus are the earliest almanacs that appeared in print.
ITALIAN: THE EARLIEST PRINTED ARITHMETIC
(1478)
128. The earliest arithmetic was printed anonymously at Treviso,
a town in Northeastern Italy. Figure 38 displays the method of solv-
ing proportions. The problem solved is as follows: A courier travels
from Rome to Venice in 7 days; another courier starts at the same
time and travels from Venice to Rome in 9 days. The distance be-
tween Rome and Venice is 250 miles. In how many days will the
1 Reproduced from Karl Falkenstein, Geschichte der Buchdruckerkunst (Leip-
zig, 1840), Plate XXIV, between p. 54 and 55. A description of the almanac of
Regiomontanus is given by A. de Morgan in the Companion to the British Almanac,
for 1846, in the article, "On the Earliest Printed Almanacs," p. 18-25.
INDIVIDUAL WRITERS
97
couriers meet, and how many miles will each travel before meeting?
Near the top of Figure 38 is given the addition of 7 and 9, and the
<£<tlettto* fce§ SPfttgifte* 3of><ttttt toott Uttn^erf *
( Johannes Regioinontanas. )
FIG. 37. — "Calendar des Magister Johann von Kunspcrk (Johannes Re«io-
montanus) Nilrnberg um 1473."
division of 63 by 16, by the scratch method.1 The number of days is
3-H-. The distance traveled by the first courier is found by the pro-
1 Our photograph IB taken from the Alii dell'Accademia Pontificia de' nuovi
Lincei, Vol. XVI (Roma, 1863), p. 570.
98 A HISTORY OF MATHEMATICAL NOTATIONS
portion 7:250 = ff:x. The mode of solution is interesting. The 7 and
250 are written in the form of fractions. The two lines which cross
e qudiite migli* baucra fatto citifcluduno t» lose.
if a fe f ondo la ritgula cofu
,
i 6 partitoje r^ 50201
1 6
Sc tu t? uol fap^tt quaita tm$lia bauera fatto ci&
cbjdunoifa per b neguU'oel.^'Dicendo
£ ptimo per qucllui t>a T\oma.
i i -t
* ^^ ^ 5 o
' T" X T
it) *
*fo
-* 1
*5o
f o o
5T-<
roji
**\
*X*S °\i 40
^^tf
^-f-f
< 5 •» f of ^. * f^
/QuelhiKbe vuntoa rsoma bauera tatto tr
.1 4 o.e _£_ poi mettila riegula per
7 d co:nn-o tea Uene^ua.
« 44
FIG. 38. — From the earliest printed arithmetic, 1478
and the two horizontal lines on the right, connecting the two numer-
ators and the two denominators, respectively, indicate what numbers
INDIVIDUAL WRITERS
99
are to be multiplied together: 7X1X16 = 112; 1X250X63 = 15,750.
The multiplication of 250 and 63 is given; also the division of 15,750
£*>i it tttiwti JaJTccbe fa.t *}$.Hi
pcrluwq? modi qui fottofcritri.
4 <*|4
* * 7 i f6\
i 7 o 3 4 7 [
a. -* ^ 7 ^ 9 X* I i J 5 T 5' /
^ua-7 001^6^6) ' 5 ^ 7 y 9 }
fc?uma. 7 o o t t 6 * 2j^
o 3 67,
» i 3 5 '8 /"L
3
o
o
~ffi
*
/
FIG. 39. — Multiplications in the Treviso arithmetic; four multiplications of
56,789 by 1,234 as given on one page of the arithmetic.
100 A HISTORY OF MATHEMATICAL NOTATIONS
by 112, according to the scratch method. Similarly is solved the
proportion 9 : 250 = J jf : x. Notice that the figure 1 is dotted in the
same way as the Roman I is frequently dotted. Figure 39 represents
other examples of multiplication.1
FRENCH: NICOLAS CHUQUET
(1484)
129. Over a century after Oresme, another manuscript of even
greater originality in matters of algebraic notation was prepared in
France, namely, Le triparly en la science des nombres (1484), by
Nicolas Chuquet, a physician in Lyons.2 There are no indications
that he had seen Oresrne's manuscripts. Unlike Oresme, he does not
use fractional exponents, but he has a notation involving integral,
zero, and negative exponents. The only possible suggestion for such
exponential notation known to us might have come to Chuquet from
the Gobar numerals, the Fihrist, and from the scholia of Neophytos
(§§ 87, 88) which are preserved in manuscript in the National Library
at Paris. Whether such connection actually existed we are not able
to state. In any case, Chuquet elaborates the exponential notation
to a completeness apparently never before dreamed of. On this sub-
ject Chuquet was about one hundred and fifty years ahead of his time;
had his work been printed at the time when it was written, it would,
no doubt, have greatly accelerated the progress of algebra. As it was,
his name was known to few mathematicians of his time.
Under the head of "Numeration," the Triparly gives the Hindu-
Arabic numerals in the inverted order usual with the Arabs:
".0.9.8.7.6.5.4.3.2.1." arid included within dots, as was customary
in late manuscripts and in early printed books. Chuquet proves
addition by "casting out the 9's," arranging the figures as follows:
5 2 >.7.
^
16 9. ' — .7.
1 Ibid., p. 550.
2 Op. cit. (public d'apres le manuscrit fonds Francois N. 1346 de la Biblio-
theque nationale de Paris et precede .(Time notice, par M. Aristide Marre),
Bullettino di Bibliog. e di Storia delle scienze mat. etfisiche, Vol. XIII (1880), p. 555-
659, 693-814; Vol. XIV, p. 413-60.
INDIVIDUAL WRITERS 101
The addition of $ and f is explained in the text, and the following
arrangement of the work is set down by itself:1
130. In treating of roots he introduces the symbol R, the first
letter in the French word ratine and in the Latin radix. A number,
say 12, he calls ratine premiere, because 12, taken as a factor once,
gives 12; 4 is a ratine seconde of 16, because 4, taken twice as a factor,
gives 16. He uses the notations #M2. equal .12., $2.16. equal .4.,
/?4.16. equal .2., #5.32. equal .2. To quote: "II conuiendroit dire
que racine piniere est entenduc pour tous nombres simples Come qui
diroit la racine premiere de .12. que Ion peult ainsi noter en mettant
.1. dessus R. en ceste maniere Rl.12. cest .12. Et #*.9. est .9. et
ainsi de tous aultres nobres. Racine seconde est celle qui posee en
deux places lune soubz laultre et puys multipliee lune par laultre pduyt
le nombre duquel elle est racine seconde Comme 4. et .4. qui multipliez
lung par laultre sont .16. ainsi la racine seconde de .16. si est .4. ... on
le peult ainsi rnettre #216. ... Et $5.32. si est .2. Racine six! se doit
ainsi mettrc I}®, et racine septiesrne ainsi I}7. ... Aultres maniercs de
racines sont que les simples devant dictes que Ion peult appeller
racines composees Come de 14. plus #2180. dont sa racine seconde si
est .3. p. #25. [i.e., 1/14+ 1/180 = 3+ 1/5] ... coe la racine seconde de
.14. p #2.180. se peult ainsi rnettre g2.14.p.g2.18Q."2
Not only have we here a well-developed notation for roots of inte-
gers, but we have also the horizontal line, drawn underneath the
binomial 14+ 'V 180, to indicate aggregation, i.e., to show that the
square root of the entire binomial is intended.
Chuquet took a position in advance of his time when he computed
with zero as if it were an actual quantity. He obtains,3 according to
his rule, z = 2± 1/4^-4 as the roots of 3x2+12-12x. He adds: "...
reste .0. Done 52.0. adioustee ou soustraicte avec .2. ou de .2. monte
.2. qui est le nob? que Ion demande."
131. Chuquet uses p and ra to designate the words plus and mains.
These abbreviations we shall encounter among Italian writers. Pro-
ceeding to the development of his exponential theory and notation,
1 Boncomp£,gni, Bullettino, Vol. XIII, p. 636.
9 Ibid. t p. 655.
3 Ibid., p. 805; Enestrom, Bibliotheca mathematica, Vol. VIII (1907-8), p. 203.
102 A HISTORY OF MATHEMATICAL NOTATIONS
he states first that a number may be considered from different points
of view.1 One is to take it without any denomination (sans aulcune
denomiaciori) , or as having the denomination 0, and mark it, say,
.12? and .13? Next a number may be considered the primary number
of a continuous quantity, called "linear number" (nombre linear), des-
ignated .121 .131 .201, etc. Third, it may be a secondary or superficial
number, such as 122. 132. 192., etc. Fourth, it may be a cubical num-
ber, such as .123. 153. I3., etc. "On les peult aussi entendre estre
nombres quartz ou quarrez de quarrez qui seront ainsi signez . 124.
184. 304., etc." This nomenclature resembles that of the Byzantine
monk Psellus of the eleventh century (§ 117).
Chuquet states that the ancients called his primary numbers
"things" (choses) and marked them .P.; the secondary numbers they
called "hundreds" and marked them .tf.; the cubical numbers they
indicated by D ; the fourth they called "hundreds of hundreds"
(champs de champ), for which the character was ttf. This ancient
nomenclature and notation he finds insufficient. He introduces a
symbolism "que Ion peult noter en ceste maniere #2.12l. #2.122. $2.123.
#2.124. etc. S3.!^. 53.122. «U2». #3.124. etc. #4.135. #<M26. etc." Here
"7i4.135." means l/13x5. He proceeds further and points out "que Ion
peult ainsi noter .12! 'm' ou moins 12.," thereby introducing the notion
of an exponent "minus one." As an alternative notation for this last
he gives ".rft.121," which, however, is not used again in this sense, but
is given another interpretation in what follows.
From what has been given thus far, the modern reader will prob-
ably be in doubt as to what the symbolism given above really means.
Chuquet's reference to the ancient names for the unknown and the
square of the unknown may have suggested the significance that he
gave to his symbols. His 122 does not mean 12X12, but our 12x2;
the exponent is written without its base. Accordingly, his ".12.1-™-"
means 12X"1. This appears the more clearly when he comes to "adi-
ouster 8! avec m.5! monte tout .3.1 Ou .10.1 avec .ra.16.1 mote tout
m.6.1," i.e., 8x-5x = 3x, 10o:--16x= -6z. Again, ".8.2 avec .12.2
montent .20.2" means 8x2+12x2 = 20x2; subtracting ".w.16?" from
".12.2" leaves "12.2 m. m. 169 qui valent autant c5me .12.2 p. 16?"2
The meaning of Chuquet's ".12?" appears from his "Example, qui
multiplie .12? par .12? montent .144. puis qui adiouste .0. avec .0.
monte 0. ainsi monte ceste multiplicacion .144?,"3 i.e., 12x°X12o:0 =
1 Boncompagni, op. cit., Vol. XIII, p. 737.
* Ibid., p. 739. 3 Ibid., p. 740.
INDIVIDUAL WRITERS 103
144x°. Evidently, £°=1; he has the correct interpretation of the ex-
ponent zero. He multiplies .12? by .10.2 and obtains 120.2; also .5.1
times .8.1 yields .40.2; .12.3 times .10.5 gives .120.8; .8.1 times .71-™-
gives .56? or .56.; .8? times .71-™- gives .56.2 Evidently algebraic
multiplication, involving the product of the coefficients and the sum of
the exponents, is a familiar process with Chuquet. Nevertheless, he
does not, in his notation, apply exponents to given numbers, i.e.,
with him "32" never means 9, it always means 3z2. He indicates
(p. 745) the division of 30 — x by x*+x in the following manner:
30. m. I1
I2 p. I1 '
As a further illustration, we give #2y.g#224.p.#214. multiplied
by #2 lj^#?_24. ra#?14. gives & 24. This is really more compact
and easier to print than our J/1I+1/24+1/1-J- times
V\\ equals 1/24 .
FRENCH: £STIENNE DE LA ROCHE
(1520)
132. Estienne de la Roche, Villefranche, published Larismethique,
at Lyon in 1520, which appeared again in a second edition at Lyon in
1538, under the revision of Gilles Huguetan. De la Roche mentions
Chuquet in two passages, but really appropriates a great deal from
his distinguished predecessor, without, however, fully entering into
his spirit and adequately comprehending the work. It is to be re-
gretted that Chuquet did not have in De la Roche an interpreter
acting with sympathy and full understanding. De la Roche mentions
the Italian Luca Pacioli.
De la Roche attracted little attention from writers antedating
the nineteenth century; he is mentioned by the sixteenth-century
French writers Buteo and Gosselin, and through Buteo by John
Wallis. He employs the notation of Chuquet, intermixed in some
cases, by other notations. He uses Chuquet's p and m for plus and
moins, also Chuquet's radical notation $2, fi3, #4, . . . . , but gives an
alternative notation: H D for #3, HI for #4, HI D for #6. His
strange uses of the geometric square are shown further by his writing
D to indicate the cube of the unknown, an old procedure mentioned
by Chuquet.
The following quotation is from the 1538 edition of De la Roche,
where, as does Chuquet, he calls the unknown and its successive
powers by the names of primary numbers, secondary numbers, etc. :
104
A HISTORY OF MATHEMATICAL NOTATIONS
"... vng chascun nombre est corisidere comme quantity continue
que aultrement on dit nombre linear qui peult etre appelle chose ou
premier: et telz nombres seront notez apposition de une unite au
dessus deulx en ceste maniere 12l ou 131, etc., ou telz nombres seront
signes dung tel characte apres eux comme 12. p. ou 13.P. ... cubes que
Ion peut ainsi marquer 12. 3 ou 13.8 et ainsi 12 D ou 13 D."1
The translation is as follows:
"And a number may be considered as a continuous quantity, in
other words, a linear number, which may be designated a thing or as
primary, and such numbers are marked by the apposition of unity
above them in this manner 12l or 131, etc., or such numbers are indi-
cated a^lso by a character after them, like 12. P, or 13. P. ... Cubes one
u,. 40. Tart <>i iol. GO/* ot IV la Koche's Larismethique ol
may mark 12.3 or 13.3 and also 12 D. or 13 D." (We have here 12X =
12x, 12.3=12x3, etc.)
A free translation of the text shown in Figure 40 is as follows:
"Next find a number such that, multiplied by its root, the product
is 10. Solution: Let the number be x. This multiplied by V x gives
V/x3= 10. Now, as one of the sides is a radical, multiply each side by
itself. You obtain z3= 100. Solve. There results the cube root of 100,
i.e., 1^100 is the required number. Now, to prove this, multiply
flOO by 1/100. But first express I^IOO as i/ , by multiplying
100 by itself, and you have 1^10,000. This multiplied by V/100 gives
V7 1,000,000, which is the square root of the cube root, or the cube
root of the square root, or 1/1,000,000. Extracting the square root
gives 1^1,000 which is 10, or reducing by the extraction of the cube
root gives the square root of 100, which is 10, as before."
1 See an article by Terquem in the Nouvelles annales de math&matiques (Ter-
quem et Gerono), Vol. VI (1847), p. 41, from which this quotation is taken. For
extracts from the 1520 edition, see Boncompagni, op. tit., Vol. XIV (1881), p. 423.
INDIVIDUAL WRITERS
105
The end of the solution of the problem shown in Figure 41 is in
modern symbols as follows:
first 1/34+7 .
x x+l x+4
2-Jx
[i.e., x = 1/34+4]
-T:
1% i ** F.
'
'" " ^
8x+lS
4 16_
16 1/34+4 second.
iiviifv*i'w^^^'fw|n^':^''^^^p^'TO(jffT»fiT|T^ff i^'i K , ^v',,1
$i^^P»^«Wr, 'v;'^^^^,^
^tey::i^^^7ii§te^
«~jyi4«"-w"^^ ,. MM v- ,,1'i-**M-.ltT*lf*_*JLJiiJ
I «». • , • -, - - $Fw^w&yf^*A
TkfrEWSF*''1"* :^^fc%
ce, . ''• .,.) !•!'.??*.%
i , , ,' i • u ^| ^^-i1^,,1 'riiHlr|i^Tl'-r- <M
jj^p_ , ,; / j / ,,_XJ^'' w/-*Lu. j.U^'/^jfc'/J^J**^1
FIG. 41.— Part of fol. 66 of De la Roche's Larismethique of 1520
ITALIAN: PIETRO BORGI (OR BORGHI)
(1484, 1488)
133. Pietro Borgi's Arithmetica was first printed in Venice in 1484;
we use the edition of 1488. The book contains no algebra. It displays
the scratch method of division and the use of dashes in operating with
fractions (§§ 223, 278). We find in this early printed Arithmetica the
use of curved lines in the solution of problems in alligation. Such
graphic aids became frequent in the solution of the indeterminate
problems of alligation, as presented in arithmetics. Pietro Borgi,
on the unnumbered folio 79B, solves the following problem: Five
sorts of spirits, worth per ster, respectively, 44, 48, 52, 60, 66 Midi,
106 A HISTORY OF MATHEMATICAL NOTATIONS
are to be mixed so as to obtain 50 ster, worth each 56 soldi. He solves
this by taking the qualities of wine in pairs, always one quality
dearer and the other cheaper than the mixture, as indicated by the
curves in the example.
16
4
10 4 10 8 12
Then 56-44 = 12; 66-56 = 10; write 12 above 66 and 10 above 44.
Proceed similarly with the pairs 48 and 60, 52 and 66. This done, add
10, 4, 10, 8, 16. Their sum is 48, but should be 50. Hence multiply
each by -J--JJ- and you obtain 10t\ as the number of ster of wine worth
44 soldi to be put into the mixture, etc*
ITALIAN: LUCA PACIOLI
(1494, 1523)
134. Introduction. — Luca Pacioli's Summa de arithmetica geo-
metria proportioni et proportionalita (Venice, 1494) l is historically
important because in the first half of the sixteenth century it served
in Italy as the common introduction to mathematics and its influence
extended to other European countries as well. The second edition
(1523) is a posthumous publication and differs from the first edition
1 Cosmo Gordon ("Books on Accountancy, 1494-1600," Transactions of the
Bibliographical Society [London], Vol. XIII, p. 148) makes the following remarks on
the edition of 1494: "The Summa de arithmetica occurs in two states. In the first
the body of the text is printed in Proctor's type 8, a medium-sized gothic. On sig.
a 1, on which the text begins, there is the broad wood-cut border and portrait-
initial L already described. In the second state of the Summa, of which the copy
in the British Museum is an example, not only do the wood-cut border and initial
disappear from a 1, but sigs. a-c with the two outside leaves of sigs. d and e, and
the outside leaf of sig. a, are printed in Proctor's type 10**, a type not observed by
him in any other book from Paganino's press. There are no changes in the text of
the reprinted pages, but that they are reprinted is clear from the fact that incorrect
head-lines are usually corrected, and that the type of the remaining pages in copies
which contain the reprints shows signs of longer use than in copies where the text
type does not vary. It may be supposed that a certain number of the sheets of the
signatures in question were accidentally destroyed, and that type 8 was already in
use. The sheets had, therefore, to be supplied in the nearest available type." The
copy of the 1494 edition in the Library of the University of California exhibits
the type 10.
INDIVIDUAL WRITERS 107
only in the spelling of some of the words. References to the number of
the folio apply to both editions.
In the Summa the words "plus" and "minus," in Italian piu and
meno, are indicated by p and m. The unknown quantity was called
"thing," in the Italian cosa, and from this word were derived in
Germany and England the words Coss and "cossic art," which in the
sixteenth and seventeenth centuries were synonymous with "algebra."
As pointed out more fully later, co. (cosa) meant our x; ce. (censo)
meant our x2; cu. (cubo) meant our x3. Pacioli used the letter 1} for
radix. Censo is from the Latin census used by Leonardo of Pisa and
Regiomontanus. Leonardo of Pisa used also the word res ("thing").
135. Different uses of the symbol ft. — The most common use of ft,
the abbreviation for the word radix or radiciy was to indicate roots.
Pacioli employs for the same purpose the small letter >vl, sometimes
in the running text,1 but more frequently when he is pressed for space
in exhibiting algebraic processes on the margin.2 He writes in Part I
of his Summa:
(Fol. 70£) 5.200. for 1/200
(Fol. 119B) ft .cuba. de .64. for
(Fol. 182A)* R.relato. for fifth root
(Fol. 182A) ft ft ft. cuba. for seventh root
(Fol. 86A) ft .6.7n.ft.2. for 1/6-1/2
(Fol. 131A) ft ft.120. for 1/120
(Fol. 182A) ft. cuba. de ft. cuba. for sixth root
(Fol. 1824) ft ft. cuba. de ft. cuba. for eighth root.
The use of the fty. for the designation of the roots of expressions con-
taining two or more terms is shown in the following example:
(Fol. 149A) &v. ft.20Jr.m4. for 1/1/20} -
The following are probably errors in the use of fty.:
(Fol. 93A) fty. 50000.rn.200. for 1/50,000-200 ,
(Fol. 93A) R Rv. 50000.m.200. for ^50,000-200 .
In combining symbols to express the higher roots, Pacioli uses the
additive principle of Diophantus, while in expressing the higher powers
1 Part I (1523), fol. 86 A.
*/Wd., fol. 124^1.
3 On the early uses of radix relata and primo relaio see Enestrom, Bibliolhcca
mathematica, Vol. XI (1910-11), p. 353.
108 A HISTORY OF MATHEMATICAL NOTATIONS
he uses the multiplication principle of the Hindus. Thus Pacioli
indicates the seventh root by R R R. cuba. (2+2+3), but the eighth
power by ce.cexe. (2X2X2). For the fifth, seventh, and eleventh
powers, which are indicated by prime numbers, the multiplication
principle became inapplicable. In that case he followed the notation
of wide prevalence at that time and later: p?r? (primo relato) for the
fifth power, 2?r? (secundo relato) for the seventh power, 3?r? (terzo
relato) for the eleventh power.1 Whenever the additive principle was
used in marking powers or roots, these special symbols became super-
fluous. Curiously, Pacioli applies the additive principle in his nota-
tion for roots, yet does not write R.I} cuba (2+3) for the fifth root,
but I}, relata. However, the seventh root he writes R R R. cuba
(2+2+3) and not 722?r?.2
136. In other parts of Pacioli's Sumrna the sign 1} is assigned alto-
gether different meanings. Apparently, his aim was to describe the
various notations of his day, in order that readers might select the
symbols which they happened to prefer. Referring to the prevailing
diversity, he says, "tante terre: tantc vsanzo."3 Some historians have
noted only part of Pacioli's uses of R, while others have given a fuller
account but have fallen into the fatal error of interpreting certain
powers as being roots. Thus far no one has explained all the uses of the
sign 1} in Pacioli's Summa. It was Julius Roy Pastor arid Gustav
Enestrom who briefly pointed out an inaccuracy in Moritz Cantor,
when he states that Pacioli indicated by R 30 the thirtieth root,
when Pacioli really designated by R .30? the twenty-ninth power. This
point is correctly explained by J. Tropfke.4
We premise that Pacioli describes two notations for representing
powers of an unknown, x2, x3, . . . . , and three notations for x. The
one most commonly used by him and by several later Italian writers
of the sixteenth century employs for x, x2, x3, x4, x5, a;6, x7, . . . . , the
abbreviations co. (cosa), ce. (cemo), cu. .(cubo), ce.ce. (censo de censo),
p?r? (primo relato), ce.cu. (censo de cubo), 2?r? (secundo relato), . . . ,5
Pacipli's second notation for powers involves the Use of I}, as al-
ready indicated. He gives: R.p? (radix prima) for #°, R.2? (radix
secunda) for x, R.3? (radix terza) for x2, . . . . , #.30? (nono relato)
for x29.8 When Enestrom asserts that folio &tB deals, not with roots,
1 Part I, fol. 67£.
2 Ibid., fol. 1824.
8 Ibid., fol. Q7B. 4 Op. cit. (2d ed.), Vol. II (1921), p, 109.
6 Op. cit., Part I, fol. 67£. « Ibid.
INDIVIDUAL WRITERS 109
out exclusively with the powers x°, x, #2, . . . . , x29, he is not quite
accurate, for besides the foregoing symbols placed on the margin of
the page, he gives on the margin also the following: "Rx. Radici;
R R. Radici de Radici; Rv. Radici vniuersale. Ouer radici legata. 0
voi dire radici vnita; R. cu. Radici cuba; $? quantita." These ex-
pressions are used by Pacioli in dealing with roots as well as with
powers, except that Rv. is employed with roots only; as we have seen,
it signifies the root of a binomial or polynomial. In the foregoing two
ases of 5, how did Pacioli distinguish between roots and powers? The
Drdinal number, pn'wa, secunda, terza, etc., placed after the 5, always
signifies a "power," or a dignita. If a root was intended, the number
effected was written after the 5; for example, 5.200. for 1/200. In
•olio 143AB Pacioli dwells more fully on the use of R in the designa-
tion of powers and explains the multiplication of such expressions as
R. 5? via. R. ll?/a R. 15a, i.e., x*Xxw = xu. In this notation one looks
in vain for indications of the exponential concepts and recognition of
:.he simple formula am*an = am+n. Pacioli's results are in accordance
with the formula am«an = am+n~1. The ordinal numbers in R 11°, etc.,
exceed by unity the power they represent. This clumsy designation
:nade it seem necessary to Pacioli to prepare a table of products,
occupying one and one-half pages, and containing over two hundred
md sixty entries; the tables give the various combinations of factors
whose products do not exceed x29. While Enestrom and Rey Pastor
lave pointed out that expressions like 6.28? mark powers and not
*oots, they have failed to observe that Pacioli makes no use whatever
3f this curious notation in the working of problems. Apparently his
aim in inserting it was encyclopedia!.
137. In working examples in the second part of the Summa,
Pacioli exhibits a third use of the sign R not previously noted by
historians. There R is used to indicate powers of numbers, but in a
:nanner different from the notation just explained. We quote from
the Summa a passage1 in which R refers to powers as well as to roots.
Wliich is meant appears from the mode of phrasing: "... 5.108. e
}uesto mca con laxis ch' 5.16. fa. 5-1728 piglia el .£. cioe recca .3. a. 5.
•a .9. parti .1728 in. 9. neuien. 192. e. 5J-92 " (.'. 1/108 and mul-
tiplying this with the axis which is 1/16 gives 1/1,728. Take {-, i.e.,
rising 3 to the second power gives 9; dividing 1,728 by 9 gives 192,
ind the 1/192. . . . .) Here "recca. 3. a. 5. fa. 9." identifies 5 with
i power. In Part I, folio 186A, one reads, "quando fia recata prima. 1.
* Ibid., Part II, fol. 12 B.
110 A HISTORY OF MATHEMATICAL NOTATIONS
co. a. R. fa. 1. ce" ("raising the x to the second power gives a;2"). Such
phrases are frequent as, Part II, folio 72B, "reca. 2. a. R. cu. fa. 8"
("raise 2 to the third power; it gives 8"). Observe that R. cu. means
the "third" power, while1 R. 3? and R. terza. refer to the "second"
power. The expression of powers by the Diophantine additive plan
(2+3) is exhibited in "reca. 3. a. R R. cuba fa. 729" ("raise 3 to
the fifth power; it gives 729").2
A fourth use of R is to mark the unknown x. We have previously
noted Pacioli's designation of x by co. (cosd) and by R. 2?. In Part II,
folio 155, he gives another way: "la mita dun censo e .12. dramme:
sonno equali a .5.R. E questo come a dire .10. radici sonno equali a vn
censo e. 24. dramme" ("Half of x2 and the number 12 are equal to 5x;
and this amounts to saying Wx are equal to x2 and the number 24").
In Part I, folio GO/?, the sign R appears on the margin twice in a
fifth role, namely, as the abbreviation for rotto ("fraction"), but this
use is isolated. From what we have stated it is evident that Pacioli
employed R in five different ways; the reader was obliged to watch his
step, not to get into entanglements.
138. Sign of equality. — Another point not previously noted by
historians is that Pacioli used the dash ( — ) as a symbol for equality.
In Part I, folio 91 Aj he gives on the margin algebraic expressions relat-
ing to a problem that is fully explained in the body of the page. We
copy the marginal notes and give the modern equivalents:
Summa (Part I, fol. 91A) Modern Equivalents
p? 1. co. m. 1. p? 1st x — y
3? 1. co. p. 1. $? " 3d x +y
1. co. m. 1. ce. de. #5? _ 36 z2-?/2 =
Rv. 1. ce. m 36 _ L ce. de ^
Valor quantitatis. the value of y .
p? 1. co, m Rv. 1. ce. m 36 1st Z- T/x2-36
2? 6 2d 6
3? 1. co. p Rv. 1. ce. m 36 . 3d x+V/x2~36
2. co. p. 6. 216 2x+6 =216
2. co. 210 2x =210
Valor rei. 105 Value of a; 105
1 Part I, fol. 67£. 2 Part II, fol. 72B.
INDIVIDUAL WRITERS 111
Notice that the co. in the third expression should be ce., and that the
.1. ce. de $&a in the fourth expression should be .1. co. de #K Here, the
short lines or dashes express equality. Against the validity of this
interpretation it may be argued that Pacioli uses the dash for several
different purposes. The long lines above are drawn to separate the
sum or product from the parts which are added or multiplied. The
short line or dash occurs merely as a separator in expressions like
Simplices Quadrata
3 _ 9
in Part I, folio 39A. The dash is used in Part I, folio 54 B, to indicate
multiplication, as in
14 15
where the dash between 5 and 7 expresses 5X7, one slanting line
means 2X7, the other slanting line 5X3. In Part II, folio 37.A, the
dash represents some line in a geometrical figure; thus d 3 fc means
that the line dk in a complicated figure is 3 units long. The fact that
Pacioli uses the dash for several distinct purposes does not invalidate
the statement that one of those purposes was to express equality. This
interpretation establishes continuity of notation between writers pre-
ceding and following Pacioli. Regiomontanus,1 in his correspondence
with Giovanni Bianchini and others, sometimes used a dash for equal-
ity. After Pacioli, Francesco Ghaligai, in his Pralica d'arithmetica, used
the dash for the same purpose. Professor E. Bortolotti informs me that
a manuscript in the Library of the University of Bologna, probaby
written between 1550 and 1568, contains two parallel dashes (=) as a
symbol of equality. The use of two dashes was prompted, no doubt,
by the desire to remove ambiguity arising from the different interpre-
tations of the single dash.
Notice in Figure 42 the word cosa for the unknown number, and
its abbreviation, co.; censo for the square of the unknown, and its con-
traction, ce.; cubo for the cube of the unknown; also .p. for "plus"
and .ra. for "minus." The explanation given here of the use of cosa,
censo, cubo, is not without interest.
1 See Maximilian Curtze, Urkunden zur Geschichte der Malhematik im Mittel-
alter und der Renaissance (Leipzig, 1902), p. 278.
112
A HISTORY OF MATHEMATICAL NOTATIONS
The first part of the extract shown In Figure 43 gives J/1/40+6+
^1/40—6 and the squaring of it. The second part gives ^1/20+2
+1/1/20—2 and the squaring of it; the simplified result is given as
1/80+4, but it should be 1/80+8. Remarkable in this second example
is the omission of the v to express vniversale. From the computation
as well as from the explanation of the text it appears that the first B
was intended to express universal root, i.e., */ 1/20+2 and not
1/20+2.
FIG. 42. — Part of a page in Luca Pacioli's Summa, Part I (1523), fol.
ITALIAN: F. GHALIGAI
(1521, 1548, 1552)
139. Ghaligai's Pratica d'arithmetica1 appeared in earlier editions,
which we have not seen, in 1521 and 1548. The three editions do not
differ from one another according to Riccardi's Biblioteca matematica
italiana (I, 500-502). Ghaligai writes (fol.
=cubo=| | | ,
xr=pronico =
ll = tronico-
— r-j, xn—dromico= -- . He uses the m° for "minus' :
and the $ and 6 for "plus," but frequently writes in full piu and meno.
1 Pratica d'ariihmetica di Francesco Ghaligai Florentine (Nuouamente Riuista,
& con somma Diligenza Ristampata. In Firenze. M.D.LlI).
INDIVIDUAL WRITERS 113
Equality is expressed by dashes ( ) ; a single dash (— ) is used
also to separate factors. The repetition of a symbol, simply to fill up
an interval, is found much later also in connection with the sign of
equality ( = ). Thus, John Wallis, in his Mathesis universal.™ ([Oxford,
1657], p. 104) writes: 1+2-3= - -0.
£mea potS rationale t mediV
el:t?5/iftfton, iRadijc qnti bino*
ifnea potes rprwle z irronafe*
iRadijc qm'ntt binomij*
2 O 2 kflgjk 2 O.
FIG. 43.— Printed on the margin of fol. 123/i of Pacioli's Summa, Part I
(1523). The same occurs in the edition of 1494.
Ghaligai does not claim these symbols as his invention, but
ascribes them to his teacher, Giovanni del Sodo, in the statement
(folio 71JB): "Dimostratione di 8 figure, le quale Giovanni del Sodo
pratica la sua Arciba & perche in parte terro 'el suo stile le dimos-
treto.' "l The page shown (Fig. 45) contains the closing part of the
I0p. tit. (1552), fols. 2B, 65; Encstrom, Bibliolhcca mathcmatica, 3. S., Vol.
VIII, 1907-8, p. 90.
114
A HISTORY OF MATHEMATICAL NOTATIONS
solution of the problem to find three numbers, P, S, T, in continued
proportion, such that S* = P+T, and, each number being multiplied
!'
j
, , r*»-v r^ Tft^rj* , * 'i if^'V^', ifrl^vt/ste^1*^ ' 4' ,vVj4 irlfTC*^'. , ,, «?, j,
--^feKfi^ ; •
l:;?l'^|f^^ ;•.'; ;
fMriotuleitit
** rUt*"*)"*™* »H*HfW^lf Y*™~ TPfpj H^ "—"'^jr, ,T= -T "' -~ »rT' T" - f ' ' | ^| ,J | 1^,1* „ | [ 1 - r
* ' ' • ' "" ''-l " ! " J'"a
'ia'^^tife^i^ttffl^i1^
ft&ft^tt£^
^.;)/-;. ^ '•",; •;';:;>cQ«^B!*l4tew^f^
dttl^oMHatotiaaa0;^^
t lts-IMr
t^l,
j^1'- tfci * '«rt jp*i|' ;l« |
FIG. 44.— Part of fol. 72 of Ghaligai's Praticad'arithmetica (1552). This exhib-
its more fully his designation of powers.
INDIVIDUAL WRITERS 115
by the sum of the other two, the sum of these products is equal to
twice the second number multiplied by the sum of the other two,
plus 72. Ghaligai lets S = 3co or 3x. He has found x = 2, and the root
of x2 equal to 1/4 .
The translation of the text in Figure 45 is as follows: "equal to
1/4, and the 'I/a;4 is equal to 1/16, hence the first quantity was
18-1/288, and the second was 6, and the third 18+1/288.
S. 3x P. and T. 9x2 18o;2+6x,X3x .
P.
T.
P. 4 ?2x2
\ — 4 --- / Value of x which is 2
18 1/324 P. was 18-1/288
36 S. was 6
1/288 T. was 18+1/288
Proof
24+1/288 24-1/288
18+1/288
432+1/93,312-288 432+1/165,888-288
288 - 1/165,888 288 - l/'9pT2
144 144
144+1/93,312^ 18-1/288
-1/165J888. 18+1/288
+ 1/165,*
144-1/93,312 =36,X6
Gives 288 216,X2
216
432
Gives 504 72
As it should 504."
116
A HISTORY OF MATHEMATICAL NOTATIONS
ffl^ •' 'i " Li£ *"»«*>-*»* **** *#**W» , ',' '' '' ' "^
' ' —7»n* 1
pdi4
9Sil^lC»itf*w^W'W^^-«^>,T,~,~ J"
^fei..^''^'.''^" '';;•' •'i^"l,ii,"i ''C';1;*"'-:- •;l'4.-",'-w-l - *•".
f?;,SjE?r ?v!;t:^^^^^f,n??:a» -
'"
FIG. 45,— Ghaligai's Praticad'arithmtica (1552), fol. 108
INDIVIDUAL WRITERS 117
The following equations are taken from the same edition of 1552:
Translation
(Folio 110) i D di D m \ di D — 1 D {x4-\x2 = x*
i D di D — 1.1 di D \x*=l\x*
in -ii-fl iz2=u
(Folio 113) iD D w 4 D 4 D }a;4-4a;2 = 4z2
i a a 8 a }z4=8z2
Ghaligai uses his combinations of little squares to mark the orders
of roots. Thus, folio 84#, # D di 3600 — die e 60, i.e., 1/3,600 = 60;
folio 727?, laH\m di 8 ditiamo 2, i.e., f/8 = 2; folio 737?, 7£ | j di 7776
for f 7,776; folio 73B, 7£ rT I di D di 262144 for \/ 262,144.
ITALIAN: HIERONYMO CARDAN
(1539, 1545, 1570)
140. Cardan uses p and m for "plus" and "minus" and $ for
"root." In his Pradica arithmeticae generalis (Milano, 1539) he uses
Pacioli's symbols nu.y co., ce., cu., and denotes the successive higher
powers, cc.ce., Rel. p., cu.ce., Rel. 2., ce.ce.ce., cu.cu., ce. Rel.1 How-
ever, in his Ars magnet (1545) Cardan does not use co. for x, ce. for #2,
etc., but speaks of "rem ignotam, quain vocamus positionem,"2 and
writes 60+20z=100 thus: "60. p. 20. positionibus acqualia 100."
Farther on3 he writes x2+2o; = 48 in the form "1. quad. p. 2. pos. aeq.
48.," x4 in the form4 "1. quadr. quad.," z5+6z3 = 80 in the form5
"r. pm p. 6. cub. 80," z5 = 7z2+4 in the form6 "rmpm 7.
quad. p. 4." Observe that in the last two equations there is a blank
space where we write the sign of equality ( = ). These equations ap-
pear in the text in separate lines; in the explanatory text is given
aequale or aequatur. For the representation of a second unknown he
follows Pacioli in using the word quantitas, which he abbreviates to
quan. or qua. Thus7 he writes 7x+37/ = 122 in the form "7. pos. p. 3.
qua. aequal. 122."
Attention should be called to the fact that in place of the p and m,
given in Cardan's Opera, Volume IV (printed in 1663), one finds in
Cardan's original publication of the Ars rnagna (1545) the signs p:
1 Ilieronymi Cardani operum tomvs quartvs (Lvgdvni, 1663), p. 14.
2 Ibid., p. 227.
*Ibid., p. 231. *Ibid.
4 Ibid., p. 237. 6 Ibid., p. 239.
7 Ars rnagna in Operum tomvs quartvs, p. 241, 242.
118 A HISTORY OF MATHEMATICAL NOTATIONS
and m:. For example, in 1545 one finds (5+ 1/— 15) (5 — I/— 15) =
25— (—15) =40 printed in this form:
" 5p: R m: 15
5m: & m: 15
25w:m: 15 qd est 40 ,"
while in 1663 the same passage appears in the form:
" 5. p. I}, m. 15.
5. m. I}, m. 15.
25. m. m. 15. quad, est 40. ",*
141. Cardan uses 1} to mark square root. He/ employs2 Pacioli's
radix vniversalis to binomials and polynomials, thus "R.V.7. p R. 4. vel
sic (R) I3.p R. 9." for 1/7+ 1/4 or 1/13+V 9; "#.7.10.p.#.16.p.3.p
8.64." for V 10 +1/16+3 +1/64. Cardan proceeds to new nota-
tions. He introduces the radix ligata to express the roots of each of
the terms of a binomial; he writes: "LR. 7. pR. 10."3 for 1/7+
1/10. This L would seem superfluous, but was introduced to dis-
tinguish between the foregoing form and the radix distincta, as in
"R.D. 9 p. R. 4.," which signified 3 and 2 taken separately. Accord-
ingly, "#.D. 4. p. 8. 9.," multiplied into itself, gives 4+9 or 13, while
the "fi.L. 4. p. #. 9.," multiplied into it/self, gives 13+1/144 = 25.
In later passages Cardan seldom uses the radix ligata and radix dis-
tincta.
In squaring binomials involving radicals, like "R.V.L. R. 5. p. R.
1. m R.V.L. R. 5. m R. 1.," he sometimes writes the binomial a second
time, beneath the first, with the capital letter X between the two
binomials, to indicate .cross-multiplication.4 Of interest is the follow-
ing passage in the Regula aliza which Cardan brought out in 1570:
"Rp: est p: R m: quadrata nulla est iuxta usum communem" ("The
square root of a positive number is positive ; the square root of a nega-
tive number is not proper, according to the common acceptation").6
1 Sec Tropfkc, op. tit., Vol. Ill (1922), p. 134, 135.
2 Cardan, op. til., p. 14, 16, of the Practica arithmeticae of 1539.
3 Ibid., p. 16.
4 Ibid., p. 194.
5 Op. tit. (Basel, 1570), p. 15. Reference taken from Enestrom, Bibliotheca
mathematica, Vol. XIII (1912-13), p. 163.
INDIVIDUAL WRITERS
119
However, in the Ars magna1 Cardan solves the problem, to divide 10
into two parts, whose product is 40, and writes (as shown above) :
" 5. p ft. m. 15.
5. m R. m.. 15.
25 ra.w. 15. quad, cst 40 .
"tCjill'fft i'lfJPiyi* " TT^M r T1!^ '"j^""' Y'I"""'"' '• | -f * \m, t
WWtfi..* •" .ffiKL W'l&J'V* ik!l:£. »..'4cn£fc4» , . ' ' •
.
FIG. 46. — ^Part of a page (255) from the Ars magna as reprinted in H. Cardan's
Operum tomvs quartvs (Lvgdvni, 1663). The Ars magna was first published in 1545.
1 Operum tomvs quartvs, p. 287.
120 A HISTORY OF MATHEMATICAL NOTATIONS
In one place Cardan not only designates known numbers by
letters, but actually operates with them. He lets a and b stand for
any given numbers and then remarks that R , is the same as , -,- ,
•xi Va .
is the same as —= .
Vb
Figure 46 deals with the cubic x3+3x2 = 21. As a check, the value
of x, expressed in radicals, is substituted in the given equation.
There are two misprints. The 226 \ should be 256 \. Second, the two
lines which we have marked with a stroke on the left should be
omitted, except the m at the end. The process of substitution is un-
necessarily complicated. For compactness of notation, Cardan's
symbols rather surpass the modern symbols, as will be seen by com-
paring his passage with the following translation:
"The proof is as in the example o;3+3:c2 = 21. According to these
rules, the result is Sl + V+^-V.-l. The cube [i.e.,
#3] is made up of seven parts:
12- ^4,846j + 1/23,487,833}:- ^4,846 \ \- 1/23,487,833 \
+ ^46,041 1+ 1/271197776^950? - V 2^096,286, 117 ^
+ ^46,041 2 + 1/2,096,354 ~1 80 jl - ^2,1 19,776,950J
+ ^256^+1/65^0631 + ^256^- 1/65,063] .
"The three squares [i.e., 3x2] are composed of seven parts in this
manner:
9+ ^4,8461+1/23^87,8331
| - V 23,487,833J
- 256^+ V 65^)63 1
-^256 ^—1/65^0631
- 1^256 ?2
- ^256 \- 1/657063}.
Now, adding the three squares with the six parts in the cube, which
are equal to the general cube root, there results 21, for the required
aggregate."
1 Deregulaaliza (1570), p. 111. Quoted by Enestrom, op. cit.t Vol. VII (1906-7),
p. 387.
INDIVIDUAL WRITERS
121
In translation, Figure 47 is as follows:
"The Quaestio VIII.
"Divide 6 into three parts, in continued proportion, of which the
sum of the squares of the first and second is 4. We let the first be the
Q_y A s T i o VIII.
Fac ex 6. tres paries, in conrinua proper-
tionc, cjuarum quadrata primx & iecundae
iun<£ta jfimiil facianc 4. ponemus pnmain
i. poiitionenV, quadratum eius dt i. qua-
dratum , txiiduum igicur ad 4. ell quadra-
tum fccundx quantitatis, id til 4. m. i« qua-
dratb , huius radicem, & i poiitionem dc-
trahe ex 6 . habebis cectiatn quantitacem>
vc vidcsj ^juaicduCla prima in tcdam> ha-
i .pof. I v. $£.4. m. r. quad. \6. m. i. pof.
m. ^. v. 4, m. i . quad.
6. pof. m. i. quad. m. ty. v. 4. quad. ni. i.
quad/quad.
4. i 6. pof.m. $t. v. 4. quad* m. i. quad.
quad.
6. pof. m. 4. xquaj. $t. v. 4. quad. m. i .
quad. quad.
56. quad p. 1 6-. m. 48. pof. xquantur 4.
quad. m. i. quad. quad. -j
i . quad, quad.p. 3 1. quad. p. i $6-
lia 48. poCp« 140* \
i. quad. p. 16. p. i. quad. quad, xqua-
lia 48. pof. ,
FIG. 47. — Part of p. 297, from the Ars magrui, as reprinted in II. Cardan's
Operum tomvs quartvs (Lvgdvni, 1663).
1. position [i.e., x]; its square is 1. square [i.e., x2]. Hence 4 minus this
is the square of the second quantity, i.e., 4—1. square [i.e., 4— #2].
Subtract from 6 the square root of this and also 1. position, and you
will have the third quantity [i.e., 6— x— 1/4— x2], as you see, because
the first multiplied by the third . . . . :
122
A HISTORY OF MATHEMATICAL NOTATIONS
4 ~x2 | G-x-l/4-x2
36x2+16-48x = 4x2-
32x2+16+x4 =
Ix4+32x2+256 = 48x+240
ITALIAN: NICOLO TARTAGLIA
(1537, 1543, 1546, 1556-60)
142. Nicolo Tartaglia's first publication, of 1537, contains little
algebraic symbolism. He writes: "Radice .200. censi piu .10. cose"
for V/2(X)x2+ lOx, and "trouamo la cosa ualer Radice .200. men. 10."
for "We find x== 1/200- 10."1 In his edition of Euclid's Elements2 he
writes "ft ft ft ft" for the sixteenth root. In his Qvesiti3 of 1546 one
reads, "Sia .1. cubo de censo piu .48. equal a 14. cubi" for "Let
z«+48 = 14x3," and "la ft. cuba de .8. ualera la cosa, cioe. 2." for "The
1^8 equals x, which is 2."
More symbolism appeared ten years later. Then he used the p
and m of Pacioli to express "plus" and "minus," also the co., ce., cu.,
etc., for the powers of numbers. Sometimes his abbreviations are
less intense than those of Pacioli, as when he writes4 men instead of m,
or5 ccn instead of ce. Tartaglia uses ft for radix or "root." Thus "la
ft # di A 6 i"6 "la ft cu. di J e |,"7 "la ft rel. di ^ e V8 "la ft cen.
cu. di ,'4 6 i,"9 "la ft cu. cu. di 6 b 6 i"10 "la ft terza rel. di 2 ^ •« e |."n
143. Tartaglia writes proportion by separating the three terms
which he writes down by two slanting lines. Thus,12 he writes "9//
5//100," which means in modern notation 9:5 = 100:x. For his
occasional use of parentheses, see § 351.
1 Nova scientia (Venice, 1537), last two pages of "Libro secondo."
2 Evclide Megareme (Venice, 1569), fol. 229 (1st ed., 1543).
3 Qvesiti, ct invent iojii (Venice, 1540), fol. 132.
4 Scconda parle del general trattalo di nvm.eri, et misvri de Nicolo Tartaglia
(Venice, 1556), fol. 88£.
6 Ibid., fol. 73. 9 Ibid., fol. 47#.
c Ibid., fol. 38. 10 76?:^., fol. 60.
7 Ibid., fol. 34. » Ibid., fol. 68.
8 Ibid., fol. 43. 12 Ibid., fol. 162.
INDIVIDUAL WRITERS
123
On the margin of the page shown in Figure 48 are given the sym-
bols of powers of the unknown number, viz., co., ce., etc., up to the
twenty-ninth power. In the illustrations of multiplication, the
absolute number 5 is marked "5w/0"> the 0 after the solidus indi-
cates the dignitd or power 0, as shown in the marginal table. His
i'r
11
i.^J!L.,J! JL ft^™^» .1 tt.-. JjiiL..- _„ J«L_ joUukJ., i,t.J,LiT Ti ' , 14-1"-— TI
«^mn
JrKKiK^^
^
ppiv^wi —<— rnrriT^ - « ',
SssttB^ms'Kurtb'^
jfff^i, f r
4;*?^»^Vi.,
«MI Jr ,*H~'i *f^1*W7"*^l7ra1^^1 WrW , Jl fj i
"''"v:*;^Sjfjs^^''--' "^
, ,/; ri lr:p,.,"h 'i' >>n
i
: ' , ;,; i [1;
"P if ^^'' ''r ^ UK * ^St 'J I J ' '/ K i'ii *' ' ' SUtfj-1 *
i~L uh'i* * ^«"ih Ji **>" ~f 1» = IA1' lAijS^«ifcjBiwJri*Jfc«*J«»'*-'>fc*fc''*R{*1L*
FIG. 48.— Part of a pa^e from Tarlaglia/s La .sr.s-/a ^arte dd general trattato de
nvmeri, et misvre (Venice, 1560), fol. 2.
illustrations stress the rule that in multiplication of one dignitd by
another, the numbers expressing the dignitd of the factors must be
added.
ITALIAN: RAPAELE BOMBELLI
(1572, 1579)
144. Bombelli's Ualgebra appeared at Venice in 1572 and again
at Bologna in 1579. He used p. and m. for "plus" and "minus."
124
A HISTORY OF MATHEMATICAL NOTATIONS
Following Cardan, Bombelli used almost always radix legata for a
root affecting only one term. To write two or more terms into one,
Bombelli wrote an L right after the # and an inverted J at the end
of the expression to be radicated. Thus he wrote: & L 7 p. ft 14 J
for our modern 1/7+1/14, also^Rg L Re L RqQ8 p.2JmRcLRq68m
2 JJ for the modern l/{^(l/68+2) - ^(1/68-2) }.
' I,;,
' "^l'lL' h:'"
!««*«f^
»' jJ(«Bidit.^k»'i*,iM*iiitl»>ttt«Wi*i*i«,li ll
' ' 71',1' '^ '''' ' ' - ' ;''; '^ "i1 J ! '
|^ ; "i|F| ,h ;j' ^ , ^,^^1^^^^ ,Jr , ^ ,^ ^r
i'"^ ' ": '" '''"" ' i ' ' * »
^ ; ,i-.
a f > ; ,;;'
», ' y1 , - ', it c«* t * «*« *_
- ' ,.
wfett IfMfW t 9 Ha*-
ir
'"' h rL-^^iSE^2to£^^«^
r,;?;,;,;' '„ :,' i-1 '' '
•t(_1' ,,;'. v ' ' [ r _ ~£f
FIG. 49. — Part of a page from Tartaglia's La sesta partc del general trattalo de
nvmeri, et misvre (Venice, 1560), fol. 4. Shows multiplication of binomials. Ob-
serve the fancy .p. for "plus." For "minus" he writes here me or men.
An important change in notation was made for the expression of
powers which was new in Italian algebras. The change is along the
line of what is found in Chuquet's manuscript of 1484. It is nothing
less than the introduction of positive integral exponents, but without
writing the base to which they belonged. As long as the exponents
were applied only to the unknown x, there seemed no need of writing
the x. The notation is shown in Figure 50.
* Copied by Cantor, op. dt., Vol. II (2d ed., 1913), p. 624, from Bombelli's
L'algebra, p. 99.
INDIVIDUAL WRITERS 125
: 4: -":-\,:--i ->x- N. 'm^
,
'•
refti > e
,
"
:'
11 4 £ |K 4 4 &
* (
:
v .-V--H;'- ::"-.;^ -"/ ; " 4. V -
'. t * -_. .
• - Jl '"•'" "'-"*"* " - :' - i* -"•-'-"
!. **
_ •T\-:ii_;-:" - " "--: ": -^V; V ' -
m» "-10*- , '" ,
•-.-•"•". t- •• • • •-•& -'' - ' •' ^*-'::' -"j? -•
-- W - r "- '";, - " . ( <^ '-•-" %*-r- " ' -
' p» id. ""-"V "
.' '- ' .. ^ • - •;;."- 'ft- - f =: . -
= • -. - ,: ;•.' . • . ; ^r t.- ;^_ '-- • .- . '
". ;,/ • '-•• •; .• . ' " -
- " ..' '.-/-;-_ . -"- :-';. ' -.. --'_-: ;;-.-f :•;;-'*- ?v.\ '-,-'',-'-• ~
9. - "- " -^" ;: ' ^: /-'-;-' - "• " : '-' • '
:.--^--^^^^::l^;^:A-^^ :
».. y ;- ^:l.-'-;'^0.:;.-"^:::-" •.;;•-.•• .
-..-. ,. : -T.:; -•;',,---•', W'V"-y;_^:^- > •'/'-. rr^x-,,^,.::^.1"-"-1 ". " ' ."-,--:',- -". ;_,
-., •• . ---.:.- -Li--^-i j- - ', : - ;-..-'-•"-.-> -."• "•-.--.,-. - te> --• •>•"*--• '': -- '•-"-•-" '• -
&^"-: :'%S-^:^ •"'-"• -; '-:""- '
-/- -..*' - ^ - . .-: • - - -•-
I* -." \ - :-"-";-; -/•-'."."-"."- :
FIG. 50.-— From Bumbelli's Lr algebra (1572)
126 A HISTORY OF MATHEMATICAL NOTATIONS
In Figure 50 the equations are:
4x2+40;,
Bombelli expressed square root by R. q.y cube root by R. c., fourth
root by R R. q., fifth root (Radice prima incomposta, ouer relata) by
R. p. r., sixth root by R. q. c., seventh root by R. s. r., the square root of
a polynomial (Radice quadrata legato) by R. q. L J ; the cube root of a
polynomial (Radice cubica legato) by R. c. L J. Some of these symbols
are shown in Figure 51. He finds the sum of ^72 — V7 1,088 and
^ V 4;352+16 to be <* 232+] ^312.
The first part of the sentence preceding page 161 of Bombelli's
Algebra, as shown in Figure 51, is "Sommisi R. c. L R. q. 4352 .p.
16.J con R. c. L 72. m. R. q. 1088.J."
145. Bombelli's Algebra existed in manuscript about twenty years
before it was published. The part of a page reproduced in Figure 52
is of interest as showing that the mode of expressing aggregation of
terms is different from the mode in the printed texts. We have here
the expression of the radicals representing x for the cubic a:3 = 32x+24.
Note the use of horizontal lines with cross-bars at the ends; the lines
are placed below the terms to be united, as was the case in Chuquet.
Observe also that here a negative number is not allowed to stand
alone: —1069 is written 0—1069. The cube root is designated by R*,
as in Chuquct.
A manuscript, kept in the Library of the University of Bologna,
contains data regarding the sign of equality ( = ). These data have
been communicated to me by Professor E. Bortolotti and tend to
show that ( = ) as a sign of equality was developed at Bologna inde-
pendently of Robert Recorde and perhaps earlier.
The problem treated in Figure 53 is to divide 900 into two parts,
one of which is the cube root of the other. The smaller part is desig-
I $« j; ,P?T: ]
:
" I
,[ r ":;; .'fV;,.;'1,;. >>'..;,;,, :;^i ^l;;l^Li^Ai|'fr
.'!^
FIG. 51. — Bombelli's Algebra, p. 161 of the 1579 impression, exhibiting the
calculus of radicals. In the third line of the computation, instead of 18,415,616
there should be 27,852,800. Notice the broken fractional lines, indicating difficulty
in printing fractions with large numerators and denominators.
128 A HISTORY OF MATHEMATICAL NOTATIONS
nated by a symbol consisting of c and a flourish (probably intended for
co) . Then follows the equation 900 fn Ico® = leu®, (our 900 - x = z3) .
One sees here a mixture of two notations for x and x3: the notation
co and cu made familiar by Luca Pacioli, and Bombelli's exponential
notation, with the 1 and 3, placed above the line, each exponent resting
in a cup. It is possible that the part of the algebra here photo-
graphed may go back as far as about 1550. The cross-writing in the
photograph begins: "in libro vecchio a carte 82: quella di far di 10
due parti: dice messer Nicolo che Pona e & 43 p 5 m RIS: et 1'altra
il resto sino a 10, cioe 5 m R 43 p. & 18." This Nicolo is supposed to
be Nicolo Tartaglia who died in 1557. The phrasing "Messer Nicolo"
implies, so Bortolotti argues, that Nicolo was a living contemporary.
If these contentions are valid, then the manuscript in question was
written in 1557 or earlier.1
I Jt
Fi(]. /)2. — From the manuscript of the Algebra of Bombelli in the Comunale
Library of Bologna. (Courtesy of Professor E. Bortolotti, of Bologna.)
The novel notations of Bombelli and of Ghaligai before him did
not find imitators in Italy. Thus, in 1581 there appeared at Brescia
the arithmetic and mensuration of Antonio Maria Visconti,2 which
follows the common notation of Pacioli, Cardan, and Tartaglia in
designating powers of the unknown.
GERMAN: IOHANN WIDMAN
(1489, 1526)
146. Widman's Behennde vnnd hubsche Rechnug auff alien Kauff-
manschafften is the earliest printed arithmetic which contains the
signs plus (+) and minus ( — ) (see §§201, 202).
1 Since the foregoing was written, E. Bortolotti has published an article, on
mathematics at Bologna in the sixteenth century, in the Periodico di Matcmaliche
(4th ser., Vol. V, 1925), p. 147-84, which contains much detailed information, and
fifteen facsimile reproductions of manuscripts exhibiting the notations then in use
at Bologna, particularly the use of a dash ( — ) and the sign ( = ) to express equality.
2 Antonii Mariae Vicecomitis Civis Placentini practica numerorum & mensu-
rarum (Brixiac, 1581).
INDIVIDUAL WRITERS
129
,kw*j '*>'-r;&;/#iV^ i
* ' '' -
r tc"t*^v •* 'rl j^v^lfP'^
?.y>%^,^^p^^^R|S|
-r-^-fe 'W^rl" :?£i?w$i$F: : / .!
"-"-."trS;',;:!,^
L- .' - "'' >' Wv*^r^
,-„"-'/_: £*$*i*^i&>*- ' ,V ^ -\ •
!%mt-,^>|' *- 1* iw 1 .;
. - . . ,
,^*1,".^| ^4^f
r • ' ' li
•N^MA4-f'.:
c i .'''V^'TIyi '* s- .i;
FKI. 53. — From a pamphlet (marked No. 59oX, in the Library of the Uni-
versity of Bologna) containing studies and notes which Professor Bortolotti con-
siders taken from the lessons of Pompeo Bolognetti ([Bologna?]-1568).
130 A HISTORY OF MATHEMATICAL NOTATIONS
4 -f T
4 - 1
3 4* Jo d>eit/6tfiiimetf
4 -- 1 p fcie3Mttn«r»nt>
3 4- 44 IfcWWfr W<«4ttff
3 4- a* — ift/*<i&ifi mi*
5 -- 1 1 H>
3 -f- 44
3 -h 19
3 — -*-«i*
-f S>
4^39^
AinUgel 14 fc. Vnb ^dijl 1 3 ma
^nt>tnad)t j i z
tt.ntmrp:td> i oo H>t)<t6tjf ctftjentttcr
21
Fia. 54. — From the 1526 edition of Widman's arithmetic. (Taken from D. E.
Smith, Rara arithmetica, p. 40.)
7<$ 15 r>
't 1^0
f W-f f
©tct ynt
4S tS 9
FIG. 55. — From the arithmetic of Grammateus (1518)
INDIVIDUAL WRITERS 131
AUSTRIAN: HEINRICH SCHREIBER (GRAMMATEUS)
(1518, 1535)
147. Grammateus published an arithmetic and algebra, entitled
Ayn new Kunstlich Buech (Vienna), printed at Niirnberg (1518), of
which the second edition appeared in 1535. Grammateus used the
fein }ti abbiren bieqttfttttttet fined ti«#
6tft.mitft:pzimrottpn>na/Teciinba
mit fecuba/teitia mir tertia jc.Tnfc maobrau*
<f>et (btyetttgen ale -)-if? tticfc:/t>nb — /
fc*r/in weUfcer fern 50 mere? en t>:ei Kf rtel»
CDanncm quantitet fiat an beybcn oiten-f-
ebcr— fofol mann
ale 9 p:i.--7U»
8 pti.— i
»— 14 tl.
3ft in bcr Sbern cju«ntitet ~j- \jnb fn bet t)ti
r rn — / vnb — |- tibertriff r-—/ fo fol Me vnber
qu«ntitec ron ber 5>bern fubtrafrtrt tceirben/ori
$ubemub:iaenfe^— f-@oaberbie wtber qua
titer if? 0r5ffcr/fo fubtraf)ir bie BIdncrn to bcr
4p?».-|-iN.
tofW. — 4N.
60 inberob0cfaRtenq»antiter wiirtfunbe
— rnb in t'cr vnbern -|-/pnb — ubertnjf t -{-/
fo Inbrrabir cn;o x^on bcm «nbcrn/»nb'5iim u>
b«0cn fdj:cib— 3(?e<54bcr/baebie rnbcrqua
ruetiibei'trijft bie £>bcrn/fo$tef)c etnd von bcm
Anbcrn/ pub ju bum cr Hen fcige — J- aid
FIG. 56. — From the arithmetic of Grammateus (1535). (Taken from D. E.
Smith, Rara arithmetica, p. 125.)
132 A HISTORY OF MATHEMATICAL NOTATIONS
plus and minus signs in a technical sense for addition and subtraction.
Figure 55 shows his mode of writing proportion: 7Glb. : 13fl. = I2lb.:x.
He finds x = 2fl. 0 s. 12] |#. [l/Z.=8s., Is. = 300].
The unknown quantity x and its powers x2, #3, . . . . , were called,
respectively, pri (primci), 2a. or se. (secondd), 3a. or ter. (terzd), 4a.
erjfci*
roitfce
fcfebm/vtifeber quoci
vnb &cr quo c tent P p;l aii
iDatimct) inultt pUct f t>ae I?mP tayl P trt ftd)/ rii
btcem
pm :4» 34. 44,
7 49-
FKJ. 57. — From the arithmetic of Grammateus (1518)
or quart, (quarto), 5a. or quit, (quinta), 6a. or sex. (sexto)] N. stands
for absolute number.
Fig. 56 shows addition of binomials. Figure 57 amounts to the
solution of a quadratic equation. In translation: "The sixth rule:
When in a proportioned number [i.e., in 1, £, x2] three quantities are
taken so that the first two added together arc equal to the third [i.e.,
INDIVIDUAL WRITERS 133
d+ex=fx*], then the first shall be divided by [the coefficient of] the
third and the quotient designated a. In the same way, divide the
[coefficient of] the second by the [coefficient of] the third and the
quotient designated 6. Then multiply the half of 6 into itself and to
the square add a; find the square root of the sum and add that to
half of 6. Thus is found the N. of 1 pri. [i.e., the value of x]. Place
the number successively in the seven-fold proportion
N: x x2 x3 z4 x5
1. 7 49 343. 2,401. 16,807.
Now I equate 12x+24 with 2}!Jz2. Proceed thus: Divide 24 by
2-J-jjx2; there is obtained 10§a. Divide also 12x by 2Jgx2; thus arises
5jJ6. Multiplying the half of b by itself gives VsVi to which adding a,
i.e., 10JJ, will yield %|8, the square root of which is ]£; add this to
half of the part 6 or -f jj, and there results the number 7 as the number
1 pri. [i.e., x]."
The following example is quoted from Grammatcus by Treutlein:1
" Gpri. + SN. Modern Symbols
Durch 6z +8
5 pri. — 7 N. 5z — 7
30 se.+40 pri. 30z2+40x
- 42 pri.- 56 N. -42z-56
30 se. - 2 pri. -56N." 30x2 - 2x - 56 .
In the notation of Grammateus, 9 /er.+30 se. — 6 pri. +48N.
stands for 9z3+30z2-6z+48.2
We see in Grammateus an attempt to discard the old cossic sym-
bols for the powers of the unknown quantity and to substitute in
their place a more suitable symbolism. The words prima, seconda, etc.,
remind one of the nomenclature in Chuquet. His notation was
adopted by Gielis van der Hoecke.
GERMAN : CHRISTOFF RUDOLFF
(1525)
148. RudohTs Behend vnnd Hubsch Rechnung durch die kunst-
reichen regeln Algebre so gemeincklich die Coss genent werden (Strass-
1 P. Treutlein, Abhandlungen zur Geschichle der Mathematik, Vol. II (Leipzig,
1879), p. 39.
2 For further information on Grammateus, see C. I. Gerhardt, "Zur Ge-
schichto der Algebra in Deutschland," Monatsbericht d. k. Akademie der Wissen-
schaften zu Berlin (1867), p. 51.
134 A HISTORY OF MATHEMATICAL NOTATIONS
burg, 1525) is based on algebras that existed in manuscript (§ 203).
Figure 58 exhibits the symbols for indicating powers up to the ninth.
The symbol for cubus is simply the letter c with a final loop resembling
the letter e, but is not intended as such. What appears below the
symbols reads in translation: "Dragma or numerus is taken here as 1.
It is no number, but assigns other numbers their kind. Radix is the
fro. J£)4&m audj je erne t>on f tir«? twgm mit ctncm
r:0f nomen t>pn anfangfc* nwf $ pftr na
ct
*v jtnftqat*
furfpfifuitn
Fio. 58.— From Rudolff's Coss (1525)
side or root of a square. Zensus, the third in order, is always a square;
it arises from the multiplication of the radix into itself. Thus, when
radix means 2, then 4 is the zensus." Adam llicse assures us that these
symbols were in general use ("zcichen ader benennung Di in gemeinen
brauch teglich gehandelt werdcnn").1 They were adopted by Adam
1 Riese's Coss was found, in manuscript, in the year 1855, in the Kirchen-
und Schulbibliothek of Marienberg, Saxony ; it was printed in 1892 in the following
publication: Adam Riese, sein Lcben, seine Rechenbtictor und seine Art zurechnen.
Die Coss von Adam Riese, by Realgymnasialrektor Bruno Berlet, in Annaberg i. E.,
1892.
INDIVIDUAL WRITERS 135
Riese, Apian, Menher, and others. The addition of radicals is
shown in Figure 59. Cube root is introduced in Rudolff s Coss of
1525 as follows: "Wiirt radix cubica in diesem algorithmo bedeut
durch solchen character wv/> al8 /vw/ 8 is zu versteen radix cubica
aufs 8." ("In this algorithm the cubic root is expressed by this char-
acter AVS/> as /WA/8 is to be understood to mean the cubic root of 8.")
The fourth root Rudolff indicated by /w/ ; the reader naturally wonders
why two strokes should signify fourth root when three strokes indi-
cate cube root. It is not at once evident that the sign for the fourth
tyenvpi t>on communicant m
jte
fa:
fa: /s'|jp fa: ^9? fa: V4<>H-
-£ jtmpf *on wacuwafrt
item ^ 4 tfti / 1 j factf S fr» CP((«» i^-*- ^108
FIG. 59.— From Rudolff's Coss (1525)
root represented two successive square-root signs, thus, \/i/. This
crudeness in notation was removed by Michael Stifel, as we shall see
later.
The following example illustrates Rudolff's subtraction of frac-
tions:1
"1 #-2 12 148-lj „
~~ von Kest 12
On page 141 of his Coss, Rudolff indicates aggregation by a dot;2
i.e., the dot in "j/. 12+1/140" indicates that the expression is
v/12+l/140, and not 1/12+1/140. In Stifel sometimes a second dot
appears at the end of the expression (§ 348). Similar use of the dot
we shall find in Ludolph van Ceulen, P. A. Cataldi, and, in form of
the colon (:), in William Oughtred.
When dealing with two unknown quantities, Rudolff represented
1 Treutlein, "Die deutsche Coss," op. cit.t Vol. II, p. 40.
*G. Wertheim, Abhandlungen zur Geschichte der Mathematik, Vol. VIII
(Leipzig, 1898), p. 153.
136 A HISTORY OF MATHEMATICAL NOTATIONS
the second one by the small letter q, an abbreviation for quantita,
which Pacioli had used for the second unknown.1
Interesting at this early period is the following use of the letters
a, c, and d to represent ordinary numbers (folio Giija) : "Nim \- solchs
collects | setz es auff ein ort | dz werd von lere wegen c genennt. Dar-
nach subtrahier_das c vom a | das iibrig werd gesprochen d. Nun sag
ich dz Vc+Vd ist quadrata radix des ersten binomij." ("Take |
this sum, assume for it a position, which, being empty, is called c.
Then subtract c from a, what remains call d. Now I say that Vc+V~d
is the square root of the first binomial.")2
149. Rudolff was convinced that development of a science is de-
pendent upon its symbols. In the Preface to the second part of
Rudolff 's Coss he states: "Das bezeugen alte bucher nit vor wenig
jaren von der coss geschriben, in welchen die quantitetn, als dragma,
res, substantia etc. nit durch character, sunder durch gantz geschribne
wort dargegeben sein, vnd sunderlich in practicirung eincs yeden
cxempels die frag gesetzt, ein ding, mit solchen worten, ponatur vna
res." In translation: "This is evident from old books on algebra,
written many years ago, in which quantities are represented, not by
characters, but by words written out in full, 'drachm/ 'thing/ 'sub-
stance/ etc., and in the solution of each special example the statement
was put, 'one thing/ in such words as ponatur, una res, etc."3
In another place Rudolff says: "Lernt die zalen der coss aus-
sprechen vnnd durch ire charakter erkennen vnd schreiben."4 ("Learn
to pronounce the numbers of algebra and to recognize and write them
by their characters.")
DUTCH: GIELIS VAN DER HOECKE
(1537)
150. An early Dutch algebra was published by Gielis van der
Hoecke which appeared under the title, In arithmetica een sonderlinge
excellet boeck (Antwerp [1537]) .5 We see in this book the early appear-
1 Chr. Rudolff, Behend vnnd Hubsch Rechnung (Strassburg, 1525), fol. Rl".
Quoted by Enestrom, Bibliotheca mathematica, Vol. XI (1910-11), p. 357.
2 Quoted from Rudolff by Enestrom, ibid., Vol. X (1909-10), p. 61.
3 Quoted by Gerhardt, op. tit. (1870), p. 153. This quotation is taken from the
second part of Gerhardt's article; the first part appeared in the same publication,
for the year 1867, p. 38-54.
* Op. cit., Buch I, Kap. 5, Bl. Dijr°; quoted by Tropfke, op. cit. (2. ed.), Vol. II,
p. 7.
6 On the date of publication, see Enestrom, op. tit., Vol. VII (1906-7), p. 211;
Vol. X (1900-10), p. 87.
INDIVIDUAL WRITERS 137
ance of the plus and minus signs in Holland. As the symbols for
powers one finds here the notation of Grammateus, N., pri., se., 3a,
4a, 5a, etc., though occasionally, to fill out a space on a line, one en-
"
f I If
fffftl
0 PI«*
X
'
•
ii
mctiofotmt jAo
tolt *
•^^^^^^^^^^l
1 A jaL-j^Aty^'A'.y ''!, ^J^J'-'K'i'l 'U'^L':,*,1 ;/• 1' "'*'
^9^k^^it^^4mmm^mmm
IT ,?, ' {,,f,VL .• • . - '( 1JT .J, i '» «-*' ,..:' f ^r'S*^*' '^-''.-I'r *i,«!Mlyd ' r ^
fni
" * irr „ . ,.,
' ' !l» ' V ' '1 -»*«. ^;, ,^^» >^r^«^
iifriftw^a^wt.^ffe^i^j^-^Mit
tt^M^ ; V&'r-.J ' ' "i ' •'.< :'';:'';; ,,! •',.-'„' '"';: ' H:;vS);i=r- ••:,
,_™^'™;.:-:;;^.'|a|9ba,: •:"-; , --^ -^ :^.
FIG. 60. — From Gielis van der Hoccke's In arithmetica (1537). Multiplica-
tion of fractions by regule cos.
counters numerus, num., or nu. in place of N.; also secu. in place of se.
For pri. he uses a few times p.
The translation of matter shown in Figure 60 is as follows: "[In
order to multiply fractions simply multiply numerators by numera-
138
A HISTORY OF MATHEMATICAL NOTATIONS
tors] and denominators by denominators. Thus, if you wish to multi-
ply -j- by
j > y°u multiply 3x by 3, this gives 9#, which you write
down. Then multiply 4 by 2x2, this gives 8x2, which you write under
9# . 9
the other ^—2. Simplified this becomes ^- , the product. Second rule:
oX oX
If you wish to multiply =- by Q 10 , multiply 20 by 16 [sic] which
""
FIG. 61. — Part of a page from M. Stifel's Arithmctica intcyra (1544), fol. 235
gives 320#, then multiply 2x by 3z+12, which gives 6x2+24x. Place
this under the other obtained above a^9 , nA , this simplified gives:1
16
, the product.
3x2+12x
As radical sign Gielis van der Hoecke does not use the German
symbols of Rudolff, but the capital H of the Italians. Thus he writes
(fol. 90B) "6+#8" for Q+l/S,-"-& 32 pri." for -l/32z.
1 The numerator should be 160, the denominator 3z-f-12.
INDIVIDUAL WRITERS 139
GERMAN: MICHAEL STIFEL
(1544, 1545, 1553)
151. Figure 61 is part of a page from Michael Stifel's important
work on algebra, the Arithmetica Integra (Nurnberg, 1544). From the
ninth and the tenth lines of the text it will be seen that he uses the
same symbols as Rudolff had used to designate powers, up to and in-
cluding x9. But Stifel carries here the notation as high as a;16. As
Tropfke remarks,1 the b in the symbol 6/3 of the seventh power leads
Stifel to the happy thought of continuing the series as far as one may
choose. Following the alphabet, his Arithmetica integra (1544) gives
cp = x11, d@ = x1*, efl^x11, etc.; in the revised Coss of Rudolff (1553),
Stifel writes 93/9, £0, J)0, (#0. He was the first2 who in print dis-
carded the symbol for dragma and wrote a given number by itself.
Where Rudolff, in his Coss of 1525 wrote 4<£, Stifel, in his 1553 edition
of that book, wrote simply 4.
A multiplication from Stifel (Arithmetica integra, fol. 236i>)8 fol-
lows:
[Concluding part "63 + 87£— 6
of a problem:] 2$ — 4
In Modern Symbols
Go;2+ 8x- 6
2x*- 4
We give Stifel's treatment of the quartic equation,
63+57£+6 aequ. 5550: "Quaeritur numerus ad quern additum suum
quadratum faciat 5550. Pone igitur quod quadratum illud faciat
%. tune radix eius quadrata fac.iet IA. Et sic IA&+1A. aequabitur
1 Op. tit., Vol. II, p. 120.
2 Rudolff, Coss (1525), Signatur Hiiij (Stifel ed. [1553], p. 149); see Tropfke,
op. tit., Vol. II, p. 119, n. 651.
3 Treutlein, op. tit., p. 39.
140 A HISTORY OF MATHEMATICAL NOTATIONS
5550. Itacq IA%. aequabit 5550— IA. Facit 1A. 74. Ergo cum.
2cC+6j+57£+6, aequetur. 5550. Sequitur quod. 74. aequetur
l3 + l#+2 ..... Facit itacq. IT^.8."1
Translation:
= 5,550 .
Required the number which, when its square is added to it, gives
5,550. Accordingly, take the square, which it makes, to be A2. Then
the square root of that square is A. Then A2+A= 5,550 and A2 =
5,550-A. A becomes 74. Hence, since z4+2z3+6z2+5z+6 = 5,550,
it follows that 74 = lz2+o;+2 ..... Therefore x becomes 8."
152. When Stifel uses more than the one unknown quantity 7£>
he at first follows Cardan in using the symbol q (abbreviation for
quantita)* but later he represents the other unknown quantities by
A, B, C ..... In the last example in the book he employs five un-
knowns, 76? A, B, C, D. In the example solved in Figure 62 he repre-
sents the unknowns by 7£, A, B. The translation is as follows:
"Required three numbers in continued proportion such that the
multiplication of the [sum of] the two extremes and the difference by
which the extremes exceed the middle number gives 4,335. And the
multiplications of that same difference and the sum of all three gives
6,069.
A-\-x is the sum of the extremes,
A — x the middle number,
2 A the sum of all three,
2x the difference by which the extremes exceed the
middle. Then 2x multiplied into the sum of the extremes, i.e., in
A+x, yields 2xA+2x2 = 4,335. Then 2z multiplied into 2 A or the
sum of all make 4xA = 6,069.
"Take these two equations together. From the first it follows
_
that xA = — -^ - . But from the second it follows that IxA =
Li
6,069 „ 4,335 -2s2 6,069 , . ., , , ,
_j _ t Hence -J- — ^ - = I~~ > *or> Slnce ^cy are eQlial to one and
the same, they are equal to each other. Therefore [by reduction]
17,340 -8z2 = 12, 138, which gives x2 = 650^ and x =
1 Arithmetica Integra, fol. 307 B.
2 Ibid., Ill, vi, 252A. This reference is taken from H. Bosnians, Bibliotheca
mathematica (3d ser., 1906-7), Vol. VII, p. 66.
141
ini
IWi4»'*
jjttrt$t,jiffytt
^tf^,q*JWi
tJUftpfjtttjr.
^ap/^* -•>.'
^%^'X'1;1',:!1,
i i inW, f| f^*'«r
tM^'l i ,V H- ~ "' i (| *|J
Ju^i^mulr
-u : f -
l
c^
'
- ,-• .. , .
FIG. 62. — From Stifel's Ariihmelica Megra (1544), fol. 313
142 A HISTORY OF MATHEMATICAL NOTATIONS
"It remains to find also \A. One has [as we saw just above] IxA
r\ f
-—- — . Since these two are equal to each other, divide each by x, and
r* f
there follows -4 = -7 — . But as x = 25|, one has 4x=102, and 6,069
~
divided by 102 gives 59|. And that is what A amounts to. Since
A—XJ i.e., the middle number equals 34, and A +x, i.e., the sum of the
two extremes is 85, there arises this new problem :
"Divide 85 into two parts so that 34 is a mean proportional between
them. These are the numbers:
B, 34, 85-JS.
Since 85B-B2= 1,156, there follows #=17. And the numbers of the
example are 17, 34, 68."
Observe the absence of a sign of equality in Stifel, equality being
expressed in words or by juxtaposition of the expressions that are
equal; observe also the designation of the square of the unknown B
by the sign B%. Notice that the fractional line is very short in the case
of fractions with binomial (or polynomial) numerators — a singularity
found in other parts of the Arithmetica integra. Another oddity is
Stifel's designation of the multiplication of fractions.1 They are writ-
ten as we write ascending continued fractions. Thus
***
means "Tres quartae, duarum tertiarum, uriius septimae," i.e., £ of f
off
The example in Fig. 62 is taken from the closing part of the Arith-
metica integra where Cardan's A rs magna, particularly the solutions of
cubic and quartic equations, receive attention. Of interest is St if el's
suggestion to his readers that, in studying Cardan's Ars magna, they
should translate Cardan's algebraic statements into the German
symbolic language: "Get accustomed to transform the signs used by
him into our own. Although his signs are the older, ours are the more
commodious, at least according to my judgment."2
1 Arithmetica integra (1548), p. 7; quoted by S. Giinther, Vermischte Unter-
suchungen (Leipzig, 1876), p. 131.
2 Arithmetica integra (Niirnberg, 1544), Appendix, p. ,306. The passage, as
quoted by Tropfke, op. cit., Vol. II (2. ed.), p. 7, is as follows: "Assuescas, signa
eius, quibus ipse utitur, transfigurare ad signa nostra. Quamvis enim signa quibus
ipse utitur, uetustiora sint nostris, tainen nostra signa (meo quids iudicio) illis
sunt commodiora."
INDIVIDUAL WRITERS 143
153. Stifel rejected RudolfTs symbols for radicals of higher order
and wrote j/j for i/~~, \/cC for f~~, etc., as will be seen more fully
later.
But he adopts Rudolff s dot notation for indicating the root of a
binomial r1
1/3-12-1/56 has for its square 12+1/^6+12-
l/i6-i/g!38-i/jl38"; i.e., "V\2+ 1/6-1/12 -1/6 has for its
square 12 +l/6> 12 -1/6 -1/138 -1/138." Again:2 "Tcrtio vide,
utru i/Vi/3 12500-50 addita ad i/j-j/J 12500+50. faciat j/V
1/350000+200" ("Third, see whether ^1/12,500-50 added to
^1/12,500+50 makes 1/1/50,000+200"). The dot is employed to
indicate that the root of all the terms following is required.
154. Apparently with the aim of popularizing algebra in Germany
by giving an exposition of it in the German language, Stifel wrote in
1545 his Deutsche arithmetical in which the unknown x is expressed
by sum, x2 by "sum: sum," etc. The nature of the book is indicated
by the following equation:
"Der Algorithmic meiner deutschen Coss branch t zurn ersten
schlecht vnd ledige zale | wie der gemein Algorithmus | als da sind
12345 etc. Zuin audern braucht er die selbigen zalen vnder diesern
namen | Suma. Vnd wirt dieser nam Suma | also verzeichnet | Sum :
Als hie I 1 sum: 2 sum: 3 sum etc ..... So ich aber 2 sum: Multi-
plicir mit 3 sum : so komen rnir 6 sum: sum: Das mag ich also lesen
6 summe summarum | wie man den im Deutsche offt findet [ suma
sumarum ..... Soil ichmultipliciren6sum: sum: sum: mit 12 sum:
sum: sum: So sprich ich | 12 mal 6. macht 72 sum: sum: sum: sum:
sum sum . . . ,"4 Translation: "The algorithm of my Deutsche Coss
uses, to start with, simply the pure numbers of the ordinary algorithm,
namely, 1, 2, 3, 4, 5, etc. Besides this it uses these same numbers
under the name of summa. And this name summa is marked sum:, as
in 1 sum: 2 sum: 3 sum, etc ..... But when I multiply 2 sum: by
3 sum: I obtain 6 sum: sum:. This I may read | 6 summe summarum \
for in German one encounters often suma sumarum ..... When I am
to multiply 6 sum: sum: sum: by 12 sum: sum: sum:, I say | 12
times 6 makes 72 sum: sum: sum: sum: sum: sum: . . . ."
1 Op. cit., fol. 138a. 2 Ibid., fol. 315a.
3 Op. oil. Inhaltend. Die Hauszrechnung. Deutsche Coss. Rechnung (1545).
4 Treutlein, op. cit., Vol. II, p. 34. For a facsimile reproduction of a page of
Stifel's Deutsche arithmetica, see D. E. Smith, Rara arithmetica (1898), p. 234.
144 A HISTORY OF MATHEMATICAL NOTATIONS
The inelegance of this notation results from an effort to render the
subject easy; Stifel abandoned the notation in his later publications,
except that the repetition of factors to denote powers reappears in
1553 in his "Cossische Progress" (§ 156).
In this work of 1545 Stifel does not use the radical signs found in
his Arithmetica integra; now he uses -%_/, £/, I/, for square, cube,
and fourth root, respectively. He gives (fol. 74) the German capital
letter 2K as the sign of multiplication, arid the capital letter 2) as the
sign of division, but does not use either in the entire book.1
155. In 1553 Stifel brought out a revised edition of RudolfFs
Coss. Interesting is StifePs comparison of RudoliTs notation of
radicals with his own, as given at the end of page 134 (see Fig. 63a),
and his declaration of superiority of his own symbols. On page 135 we
read: "How much more convenient my own signs are than those of
Rudolff, no doubt everyone who deals with these algorithms will
notice for himself. But I too shall often use the sign i/ in place of the
1/j, for brevity.
"But if one places this sign before a simple number which has not
the root which the sign indicates, then from that simple number arises
a surd number.
"Now my signs are much more convenient and clearer than those
of Christoff . They are also more complete for they embrace all sorts of
numbers in the arithmetic of surds. They are [here he gives the symbols
in the middle of p. 135, shown in Fig. 636]. Such a list of surd numbers
ChristofFs symbols do not supply, yet they belong to this topic.
"Thus my signs are adapted to advance the subject by putting in
place of so many algorithms a single and correct algorithm, as we
shall see.
"In the first place, the signs (as listed) themselves indicate to
you how you are to name or pronounce the surds. Thus, j//36 means
the sursolid root of 6, etc. Moreover, they show you how they are to
be reduced, by which reduction the declared unification of many
(indeed all such) algorithms arises and is established/'
156. Stifel suggests on folio 61# also another notation (which,
however, he does not use) for the progression of powers of xt which he
calls "die Cossische Progress" We quote the following:
"Es mag aber die Cossische Progress auch also verzeychnet wer-
den:
012 3 4
1 • \A • IAA • IAAA • IAAAA • etc.
1 Cantor, op. tit,, Vol. II (2. cd., 1913), p. 444.
INDIVIDUAL WRITERS
145
auch also:
0123 4
M# • IBB • IBBB • IBBBB • etc.
Item auch also:
0123 4
1 - 1C • ICC - 1CCC • 1CCCC • etc.
Vnd so fort an von andern Buchstaben."1
.
,
FIG. 63a.~ This shows p. 134 of StifeTs edition of Rudolffs Coss (1553)
1 Treutlein, op. cit., Vol. II (1879), p. 34.
146
A HISTORY OF MATHEMATICAL NOTATIONS
We see here introduced the idea of repeating a letter to designate
powers, an idea carried out extensively by Harriot about seventy-five
/jgrufnicKrfHftivof
i.
out
FIG. 636,— This shows p. 135 of Stifel's edition of Rudolff s Coss (1553)
INDIVIDUAL WRITERS 147
years later. The product of two quantities, of which each is repre-
sented by a letter, is designated by juxtaposition.
GERMAN: NICOLAUS COPERNICUS
(1566)
157. Copernicus died in 1543. The quotation from his De revolu-
tionibus orbium coelestium (1566; 1st ed., 1543)1 shows that the exposi-
tion is devoid of algebraic symbols and is almost wholly rhetorical.
We find a curious mixture of modes of expressing numbers: Roman
numerals, Hindu-Arabic numerals, and numbers written out in words.
We quote from folio 12:
"Circulum autem communi Mathematicorum consensu in
CCCLX. partes distribuirnus. Dimetientem uero CXX. partibus
asciscebant prisci. At posteriores, ut scrupulorurn euitarent inuolu-
tionem in multiplicationibus & diuisionibus numcrorum circa ipsas
lincas, quae ut plurimum incommensurabiles sunt longitudine, saepius
etiam potentia, alij duodccies centena milia, alij uigesies, alij aliter
rationalem constituerunt diametrum, ab eo tern pore quo indicae
numerorurn figurae sunt usu receptae. Qui quidem numerus qucm-
cunque alium, sine Graccum, sine Latinum singular! quadam prompti-
tudine superat, & omni generi supputationum aptissime sese accommo-
dat. Nos quoq, earn ob causam accepimus diametri 200000. partes
tanquam sufficientes, que, possint errorern excludere paten tern."
Copernicus does not seem to have been exposed to the early move-
ments in the fields of algebra and symbolic trigonometry.
GERMAN: JOHANNES SCHEUBEL
(1545, 1551)
158. Scheubel was professor at the University of Tubingen, and
was a follower of Stifcl, though deviating somewhat from Stifcl's
notations. In ScheubeFs arithmetic2 of 1545 one finds the scratch
method in division of numbers. The book is of interest because it
docs not use the + and — signs which the author used in his algebra;
the + and — were at that time not supposed to belong to arithmetic
proper, as distinguished from algebra.
1 Nicolai Copcrnici Torinensis de Revolvlionibus Orbium Codcstium, Libri VI.
.... Item, de Libris Revolvtionvm Nicolai Copcrnici Narratio prima, per M. Georgi-
um loachimum Rheticum ad D. loan. Schonerum scripla. Basileae (date at the end
of volume, M.D.LXVI).
2 De Nvmeris el Diversis Rationibvs seu Regulis computalionum Opusculum, a
loanne Schcubelio compositum .... (1545).
148 . A HISTORY OF MATHEMATICAL NOTATIONS
Scheubel in 1550 brought out at Basel an edition of the first six
books of Euclid which contains as an introduction an exposition of
algebra,1 covering seventy-six pages, which is applied to the working
of examples illustrating geometric theorems in Euclid.
159. Scheubel begins with the explanation of the symbols for
powers employed by Rudolff and Stifel, but unlike Stifel he retains a
symbol for numerus or dragma. He explains these symbols, up to the
twelfth power, and remarks that the list may be continued indefinitely.
But there is no need, he says, of extending this unwieldy designation,
since the ordinal natural numbers afford an easy nomenclature. Then
he introduces an idea found in Chuquet, Grammateus, and others,
but does it in a less happy manner than did his predecessors. But
first let us quote from his text. After having explained the symbol for
dragma and for x he says (p. 2) : "The third of them 3, which, since it
is produced by multiplication of the radix into itself, and indeed the
first [multiplication], is called the Prima quantity and furthermore is
noted by the syllable Pri. Even so the fourth c£, since it is produced
secondly by the multiplication of that same radix by the square, i.e., by
the Prima quantity, is called the Second quantity, marked by the sylla-
ble Se. Thus the fifth sign 33, which springs thirdly from the multiplica-
tion of the radix, is called the Tertia quantity, noted by the syllable
Ter "2 And so he introduces the series of symbols, N.t Ra., Pri.,
Re., Ter., Quar., Quin., Sex., Sep , which are abbreviations for
the words numerus, radix, prima quantitas (because it arises from one
multiplication), secunda quantitas (because it arises from two multi-
plications), and so on. This scheme gives rise to the oddity of desig-
nating xn by the number n— I, such as we have not hitherto encoun-
tered. In Pacioli one finds the contrary relation, i.e., the designation
of xn~l by xn (§ 136). ScheubcPs notation does not coincide with that
of Grammateus, who more judiciously had used pri., se., etc., to desig-
nate X, x2, etc. (§ 147). ScheubePs singular notation is illustrated by
1 Evdidis Megarensis, Philosophi et Mathematici excellentissimi, sex libri
prior es de Geometrids principijs, Graed et Latini .... Alyebrae porro rcgvlae,
propter nvmerorum excmpla, passim proposition/thus adiecta, his libris praemissae
sunt, eadenque demonstratae. Author e loanne jSchcvbelio, .... Basileae (1550). I
used the copy belonging to the Library of the University of Michigan.
2 "Tertius de, g. qui cu ex multiplicatione radicis in se producatur, et primo
quidem: Prima quantitas, et Pri etiam syllaba notata, appclletur. Quartus uer6 cC
quia ex multiplicatione ciusdem radicis cum quadrato, hoc eat, cum prima quanti-
tate, secundd producitur: Se syllaba notata, Secunda quantitas dicitur. Sic
character quintus, gg, quia ex multiplicatione radicis cum secunda quantitate
tertio nascitur: Ter syllaba notata, Tertia etiam quantitias dicitur "
INDIVIDUAL WRITERS 149
Figure 64, where he shows the three rules for solving quadratic equa-
tions. The first rule deals with the solution of 4x2+3x = 217, the sec-
ond with 3x+175-4o;2, the third with 3z2+217 = 52z. These differ-
ent cases arose from the consideration of algebraic signs, it being de-
sired that the terms be so written as to appear in the positive form.
Only positive roots are found.
ALIVD EXEMPLVM.
P R I M I C A N O N I $. $ E C V N D I C A N O N 1 $.
Prf. ra, N ra, N prf.
4 -H J gquales 217 3 + 17? arqu. 4
Hie, quia maximi charafleris nttmerus non cftumtasjdiuifion^utdidhrai"
eft, ci fuccuni debct. Venfunt autem fa<3a diuifionc,
pru ra« N ra» N pnb
| in fe, f?
ucnu ^||«.
funt 7 f minus f . font <j| plus |
manent 7 ucmunt 7
radios ualon radicfs ualor.
ALIVD TERT1I CANONU
a pri, H- 217 N acquales 51 ra.
Ethic,qma maximi characflens numerusnoncftunitas,diuifioncci(uccu^
rcndum erit. Veniunc autem hoc fadto,
t pri> -t~ a|7 N requales ff N
Jf infe. *2/> minus Ij7,manec ^
fde
Hufus ra. qua, eft tf < sf, &Tmanent7,uelproue*
tad
niunt 10 1, Vtercp radio's ualor , quod examinari poteft»
FIG. 64. — 'Part of p. 28 in Sehcubel's Introduction to his Euclid, printed at
Basel in 1550.
Under proportion we quote one example (p. 41) :
" 3 ra.+4 N. ualent 8 se.+4: pri.
quanti 8 ter.—4: ra.
64 sex. +32 quin. -32 ter. - 16 se.
Foot - ~ - "
150 A HISTORY OF MATHEMATICAL NOTATIONS
In modern notation:
3z+4 are worth &r3+4;r2
how much 8x4— 4x .
64z7+32z6-32o;4- Wx*
Result - .
In the treatment of irrationals or numeri surdi Scheubel uses two
notations, one of which is the abbreviation Ra. or ra. for radix, or
"square root," ra.cu. for "cube root," ra.ra. for "fourth root." Con-
fusion from the double use of ra. (to signify "root" and also to signify
x) is avoided by the following implied understanding: If ra. is fol-
lowed by a number, the square root of that number is meant; if ra.
is preceded by a number, then ra. stands for x. Thus "8 ra." means
Sx; "ra. 12" means 1/12.
Scheubel's second mode of indicating roots is by RiidolfTs sym-
bols for square, cube, and fourth roots. He makes the following state-
ment (p. 35) which relates to the origin of j/: "Many, however, are
in the habit, as well they may, to note the desired roots by their
points with a stroke ascending on the right side, and thus they prefix
for the square root, where it is needed for any number, the sign j/:
for the cube root, AW/ ; and for the fourth root AA/-"1 Both systems
of notation are used, sometimes even in the same example. Thus, he
considers (p. 37) the addition of "ra. 15 ad ra. 17" (i.e., 1/15+1/17)
and gives the result "ra.coZ. 32+1/1020" (i.e. 1/32 +1/1, 020).
The ra.col. (radix collecti) indicates the square root of the binomial.
Scheubel uses also the ra.re (radix residui) and radix binomij. For
example (p. 55), he writes "ra.re. i/15-i/12" for 1/1/15-1/12.
Scheubel suggests a third notation for irrationals (p. 35), of which he
makes no further use, namely, radix se. for "cube root," the abbrevia-
tion for secundae quantitatis radix.
The algebraic part of ScheubePs book of 1550 was reprinted in
1551 in Paris, under the title Algebrae compendiosa facilisqve description
1 "Solent tamen multi, ct benc etiam, has desideratas radices, suis punctis
cum lines quadam a dextro latere asccndente, notare, atque sic pro radice quidern
quadrata, ubi haec in aliquo nurnero dosideratur, not am \/\ pro cubica uero,
/VV\/ : ac radicis radice deinde, /w/ praeponunt."
2 Our information on the 1551 publication is drawn from H. Staigmtiller,
"Johannes Scheubel, ein deutscher Algebraiker des XVI. Jahrhunderts," Abhand-
lungen zur Geschichte der Mathematik, Vol. IX (Leipzig, 1899), p. 431-69; A.
Witting and M. Gebhardt, Beispielc zur Geschichte der Mathematik, II. Teil
INDIVIDUAL WRITERS 151
It is of importance as representing the first appearance in France of
the symbols + and — and of some other German symbols in algebra.
Charles Hutton says of Scheubel's Algebrae compendiosa (1551):
"The work is most beautifully printed, and is a very clear though
succinct treatise; and both in the form and matter much resembles a
modern printed book."1
MALTESE: WIL. KLEBITIUS
(1565)
160. Through the courtesy of Professor H. Bosnians, of Brussels,
we are able to reproduce a page of a rare and curious little volume
containing exercises on equations of the first degree in one unknown
number, written by Wilhelrn Klebitius and printed at Antwerp in
1565.2 The symbolism follows Scheubel, particularly in the fancy
form given to the plus sign. The unknown is represented by "1R."
The first problem in Figure 65 is as follows: Find a number whose
double is as much below 30,000 as the number itself is below 20,000.
In the solution of the second and third problems the notational peculi-
arity is that J/2. — -J- is taken to mean Jfi. — -J-/i., and 1/J.—- £ to mean
IR.-iR.
GERMAN: CIIRISTOPHORUS CLAVIUS
(1608)
161. Though German, Christophorus Clavius spent the latter
part of his life in Rome and was active in the reform of the calendar.
His Algebra? marks the appearance in Italy of the German + and
— signs, and of algebraic symbols used by Stifel. Clavius is one of
the very first to use round parentheses to express aggregation. From
his Algebra we quote (p. 15): "Pleriqve auctores pro signo + ponunt
literam F, vt significet plus: pro signo vero — ponunt literam
M, vt significet minus. Bed placet nobis vti nostris signis, vt a
literis distinguantur, ne confusio oriatur." Translation: "Many
authors put in place of the sign + the letter P, which signifies "plus":
(Leipzig-Berlin, 1913), p. 25; Tropfke, op. cit., Vol. I (1902), p. 195, 198; Charles
Hutton, Tracts on Mathematical and Philosophical Subjects, Vol. II (London, 1812),
p. 241-43; L. C. Karpinski, Robert of Chester's .... Al-Khowarizmi, p. 39-41.
1 Charles Hutton, op. cit., p. 242.
2 The title is Insvlae Melitensis, qvam alias Maltam vocant, Historia, quaestionib.
aliquot Mathematicis reddita iucundior. At the bottom of the last page: "Avth.
Wil. Kebitio."
* Algebra Christophori Clavii Bambergensis e Societate lesv. (Romae,
M.DC.VIII).
152
A HISTORY OF MATHEMATICAL NOTATIONS
likewise, for the sign — • they put the letter Mf which signifies "minus."
But we prefer to use our signs; as they are different from letters, no
confusion arises/'
In his arithmetic, Clavius has a distinct notation for "fractions of
fractional numbers," but strangely he does not use it in the ordinary
.:t.
ad
H;^*p
'"rr:"'
^
,.:,,;t:*.*l'|,
' '" '
FIG. 65.-— Page from W. Klebitius (1565)
multiplication of fractions. His
means $ of
4
He says: "Vt
praedicta minutia minutiae ita scribenda est f • \ • pronuntiaturque
sic. Tres quintae quatuor septimaru vnius integri."1 Similarly,
i • 2 * a ' 1 * yields Tfy. The distinctive feature in this notation is the
1 Epitome arithmeticae (Rome, 1583), p. 68; see also p. 87.
INDIVIDUAL WRITERS 153
omission of the fractional line after the first fraction.1 The dot cannot
be considered here as the symbol of multiplication. No matter what
the operation may be, all numbers, fractional or integral, in the
C A P. XXVIII. ,5*
SferurfusBinomtumprimum 7^^2880, Maius nomcnya.
fecabitur in duas partes producentes 710* quartam partem quadrati
1880. maioris nominis , hac ratione*
Semi/Hsmaiorisnomini*7a-eil36. a «/£ 60 + J% 12
cuius quadratoxij^/detraSaquarti Jfr 60 -j- J% n
pars pratdi&a 7*0* relinquft T7*. cur , ,v _rt , t_ J
ins radix 24* addita ad femiffem uo- 6 T jj ™ T "
minaram 3 6. & detrafta a.b eadein/a* /"^ ff ..'.?. _ .
c it parces ^ux/Ttas ^o, & x s. Ergo ra-» 721 Hh Vy 18.80
dix Binomij eft JK 6o-|*^ n. cjuod
hie probatum eft per muitiplicationem radicis in (e quadrate •
Sit quoque el icienda radix exhoc refiduo fcxto JK 60 — Vfr 12;
Matus nomen Vj< 6o.diflribuetar in duas partes producetes^.quar-*
tarn parrem quadrat: i *. minoris nominisr hoc pa^to » Semiflis ma*
ions nominis 4% 60+ eft J% ij.,a emus quadrato ij.detraCla nomi-
nata pars quarta 5. relinquit u. cuius radix J# n^ addita ad fe-
miirem J% i j. prardi<flam, 5: ab ? adem fublata facit partes JK i f -f*
VK 1 2. & J% i $ — Vfr i a. Ergo radi!x difli Re/Tduf fexti eft 7
i J + J% i i>*— ^ (Vjf i J — Vfc Tz) quod He probatum eft *
if + Vjf t*) ~ ^ fJK IT — ft Ji
15 +•</* u; — Jy 6/y i? ~ ^/y n
Quadrata partium. J% ij + <!% ** &
Summa. *1% 6* — Jjf. ix
Nam quadrata pattium faciunt Jv 6*. nimirum* duplum J% i j. Et
ex vna parte Jlf (J% i J 4 Vtf i *J in alteram 7- /^y i y — J% n)
fit — <$?. quippecumqiradratumix ex quadrato ij.fubduftum
relinquat 3. cm prarponendum eft fignum /^ cum ffgno —» . pro*
£ ter Refiduum. Duplum autem-r1 ^ff J* tacit — ^/K u^
FIG. 66. — A page in Clavius' Algebra (Rome, 1608). It shows one of the very
earliest uses of round parentheses to express aggregation of terms.
arithmetic of Clavius are followed by a dot. The dot made the
numbers stand out more conspicuously.
1 In the edition of the arithmetic of Clavius that appeared at Cologne in 1601,
p. 88, 126, none of the fractional lines are omitted in the foregoing passages.
154 A HISTORY OF MATHEMATICAL NOTATIONS
As symbol of the unknown quantity Clavius uses1 the German 7£«
In case of additional unknowns, he adopts IA, IB, etc., but he refers
to the notation lq, 2q, etc., as having been used by Cardan, Nonius,
and others, to represent unknowns. He writes: 33^+4^4, 4B—3A for
Clavius' Astrolabium (Rome, 1593) and his edition of the last
nine books of Euclid (Rome, 1589) contain no algebraic symbolism
and are rhetorical in exposition.
BELGIUM: SIMON STEVIN
(1585)
162. Stevin was influenced in his notation of powers by Bombelli,
whose exponent placed in a circular arc became with Stevin an ex-
ponent inside of a circle. Stevin's systematic development of decimal
fractions is published in 1585 in a Flemish booklet, La thiende,'2 and
also in French in his La disme. In decimal fractions his exponents may
be interpreted as having the base one-tenth. Page 16 (in Fig. 67) shows
the notation of decimal fractions and the multiplication of 32.57 by
89.46, yielding the product 2913.7122. The translation is as follows:
"III. Proposition, on multiplication: Being given a decimal frac-
tion to be multiplied, and the multiplier, to find their product.
"Explanation of what is given: Let the number to be multiplied be
32.57, and the multiplier 89.46. Required, to find their product.
Process: One places the given numbers in order as shown here and
multiplies according to the ordinary procedure in the multiplication of
integral numbers, in this wise: [see the multiplication].
"Given the product (by the third problem of our Arithmetic)
29137122; now to know what this means, one adds the two last of the
given signs, one (2) and the other (2), which are together (4). We
say therefore that the sign of the last character of the product is (4),
the which being known, all the others are marked according to their
successive positions, in such a manner that 2913.7122 is the required
product. Proof: The given number to be multiplied 32.57 (according
to the third definition) is equal to 32^ ^fa, together 32-^fo. And
for the same reason the multiplier 89.46 becomes 89-^oV Multiplying
the said 32jVo by the same, gives a product (by the twelfth problem
of our Arithmetic) 2913iVoVoJ but this same value has also the said
product 2913.7122; this is therefore the correct product, which we
1 Algebra, p. 72.
2 A facsimile edition of La "thiende" was brought out in 1924 at Anvers by
H. Bosnians.
INDIVIDUAL WRITERS
155
were to prove. But let us give also the reason why © multiplied by
0, gives the product 0 (which is the sum of their numbers), also
why 0 times © gives the product ®, and why (O) times ® gives ®,
etc. We take -& and T$T (which by the third definition of this Disme
s*
are .2 and .03; their product is which, according to our third
1UUU
definition, is equal to .006. Multiplying, therefore, © by © gives the
Iff,
1,-f, 4^ i
~ r
,* . _.* 'i.rJ5Jl' ' . L /';[M,J
FIG. 67. — Two pages in S. Stevin's Thiende (1585). The same, in French, is
found in Les ceuvres mathematiques de Simon Stevin (ed. A. Girard; Leyden, 1634),
p. 209.
product ©, a number made up of the sum of the numbers of the given
signs. Conclusion: Being therefore given a decimal number as a
multiplicand, and also a multiplier, we have found their product, as
was to be done.
"Note: If the last sign of the numbers to be multiplied is not the
same as the sign of the last number of the multiplier, if, for example,
the one is 30708©, and the other 5040, one proceeds as above
156 A HISTORY OF MATHEMATICAL NOTATIONS
and the disposition of the characters in the operation is as shown:
[see process on p. 17]."
A translation of the La disme into English was brought out by
Robert Norman at London in 1608 under the title, Disme: The Art of
QVSSTION XX*
* un@td, qucfonquarre — it muttipMptrU
fotnmt du double d icdui©, &lcquarrcde— i # 4, It
CoKSr&VCTION.
Soitlenombrerequis 10 4
Son qUarrc i @, auquelajouftc — it
Qui mulripliepar la fpmmp du double du nota-
Jbre rcquis , & le quarr^ dc — z £c 4 > qtii eft
Bgalau quarrc du produidde-— i,Dar id
,
Lcfqucls rcduiiis, i ® (era cgale a — i
48 ; Et i © par Ic 71 pf oblcmc, vaudra 4.
Jc di quc 4 eft Ic nombrt f cquis. Demon ftration. Le
quarrc dc 4*eft 16, qui ivcc — u fai6l 4 , qui mulripli6
par i£ (i tf pour la fomme du double d'iceluy 4, & Ic.
quarrc de — i &: encore 4 ) faift ^4 , qui font egales ail
quarredu produiddc — i,par Ie4trouve? felon le re-
qu'il falloit deoioourer*
FIG. 68. — From p. 98 of U arithmetique in Stevin's (Euvres mathematiqucs
(I^eyden, 1634).
Tenths, or Dedmall Arithmetike. Norman does not use circles, but
round parentheses placed close together, the exponent is placed high,
as in (2). The use of parentheses instead of circles was doubtless
typographically more convenient.
Stevin uses the circles containing numerals also in algebra. Thus
INDIVIDUAL WRITERS 157
a circle with 1 inside means x, with 2 inside means x2, and so on. In
Stevin's (Euvres of 1634 the use of the circle is not always adhered to.
Occasionally one finds, for x4, for example,1 the signs (4) and (4).
The translation of Figure 68 is as follows: "To find a number such
that if its square —12, is multiplied by the sum of double that num-
ber and the square of —2 or 4, the product shall be equal to the square
of the product of —2 and the required number.
Solution
"Let the required number be x 4
Its square x2, to which is added — 12 gives x2— 12 4
This multiplied by the sum of double the re-
quired number and the square of —2 or 4, i.e.,
by 2x+8, gives 2x3+8x2- 24x- 96 equal to the 64
square of the product pf — 2 and x, i.e., equal
to. ... 4x2 Which reduced, x3 = - 2x2+12x+48;
and x, by the problem 71, becomes 4. I say
that 4 is the required number.
"Demonstration: The square of 4 is 16, which added to — 12 gives
4, which multiplied by 16 (16 being the sum of double itself 4, and
the square of —2 or 4) gives 64, which is equal to the square of the
product of —2 and 4, as required; which was to be demonstrated. "
If more than one unknown occurs, Stevin marks2 the first un-
known "1O," the second "1 secund. O," and so on. In solving a
Diophantine problem on the division of 80 into three parts, Stevin
represents the first part by "1O," the second by "1 secund. O," the
third by "-© -1 secund. ©+80." The second plus J the first + 6
minus the binomial | the second + 7 yields him "£ secund. O-f
-JO— 1." The sum of the third and J the second, + 7, minus the
binomial \ the third + 8 yields him "|O — H secund. O + 4fa-" By
the conditions of the problem, the two results are equal, and he ob-
tains "1 Secund. O Aequalem — {J JO +45." In his U arithmetique?
one finds "12 sec. ®+23@M sec. ©+10®," which means 12i/4+
23x?/2+10x2, the M signifying here "multiplication" as it had with
Stifel (§ 154). Stevin uses also D for "division."
163. For radicals Stevin uses symbols apparently suggested by
1 Les (Euvres malhematiques de Simon Stevin (1634), p. 83, 85.
2 Stevin, Tomvs Qvintvs mathematicorvm Hypomnemalvm de Miscellaneis
(Leiden, 1608), p. 516.
8 Stevin, (Euvres mathtmatiques (Leyden, 1634), p. 60, 91, of "Le II. livre
d'arith."
158 A HISTORY OF MATHEMATICAL NOTATIONS
those of Christoff Rudolff, but not identical with them. Notice the
shapes of the radicals in Figure 69. One stroke yields the usual square
root symbol j/, two strokes indicate the fourth root, three strokes
the eighth root, etc. Cube root is marked by y followed by a 3 inside
a circle; VA/ followed by a 3 inside a circle means the cube root
twice taken, i.e., the ninth root. Notice that i/^X® means 1/3 times
z2, not I/So;2 ; the X is a sign of separation of factors. In place of the
u or v to express "universal" root, Stevin uses bino ("binomial") root.
Stevin says that f placed within a circle means #*, but he does not
actually use this notation. His words are (p. 6 of (Euvres [Arithmetic]),
"•f en un circle seroit le charactere de racine quarr£e de (a), par ce
que telle f en circle multiplide en soy donne produict (5), et ainsi des
autres." A notation for fractional exponents had been suggested much
earlier by Oresme (§123).
LORRAINE: ALBERT GIRARD
(1629)
164. Girard1 uses + and — , but mentions -r as another sign used
for "minus." He uses = for "difference entre les quantitez oft il se
treuve." He introduces two new symbols: ff, plus que; §, moins que.
In further explanation he says : "Touchant les lettres de F Alphabet au
lieu des nombres: soit A & aussi B deux grandeurs: la somme est
A+B, leur difference est A=B, (ou bien si A est majeur on dira que
.A
c'est A— B) leur produit est AB, mais divisant A par B viendra -~
comme £s fractions: les voyelles se posent pour les choses incognues."
This use of the vowels to represent the unknowns is in line with the
practice of Vieta.
The marks (2), (3), (4), . . . . , indicate the second, third, fourth,
. . . . , powers. When placed before, or to the left, of a number, they
signify the respective power
[BRIEF VE COLLECTION DES of that number; when placed
CHA-RACTERES QjfON vsERA B N| after a number, they signify
CIST* AIUTHMETK^VE. the power of the unknown
-TTEuqucUcognoinincedcscharadlcrcscftacgran- quantity. In this respect
V dcconrcqucncc,parcequonl«ufccnrArithiiic- . , ,
au lieu d« mots, nous les ajoufterons icy, (com- uirard f ollows the general plan
^bicnquauprcccdcntchafcunacftcamplcmcntacciarc found in Schoner's edition of
[Continued on page i5'jj the Algebra of Ramus. But
1 Invention nouvelle en I'Algebre, A Amsterdam (M.DC.XXIX); reimpression
par Dr. D. Bierens de Haan (Leiden, 1884), fol. B.
INDIVIDUAL WRITERS
159
4tcr
Girard adopts the practice of
Stevin in using fractional ex-
ponents. Thus, "(f)49" means
(1/49)3 = 343, while "49(f)"
means 49z3. He points out
that 18(0) is the same as 18,
that (1)18 is the same as
18(0).
We see in Girard an ex-
tension of the notations of
Chuquet, Bombelli, and Ste-
vin ; the notations of Bombelli
and Stevin are only variants
of that of Chuquet.
The conflict between the
notation of roots by the use
of fractional exponents and
by the use of radical signs
had begun at the time of
Girard. "Or pource que y
est en usage, on le pourra
prendre au lieu de (-0 a cause
aussi de sa facilite, signifiant
racine seconde, ou racine
quaree; que si on veut pour-
suivre la progression on pour-
ra au lieu de j/ marquer i/ ';
& pour la racine cubique, ou
tierce, ainsi \/ ou bien (£), ou
bi6 cf, ce qui peut estre au
choix, mais pour en dire mon
opinion les fractions sont plus
expresses & plus propres a
exprimer en perfection, & j/
plus faciles et expedientes,
comme i/ 32 est a dire la ra-
cine de 32, & est 2. Quoy que
ce soit Tun & 1'autre sont f acils
a comprendre, mais |/ et c£ sont pris pour faciliteV' Girard appears to
be the first to suggest placing the index of the root in the opening of
the radical sign, as {/. Sometimes he writes j/V for \/.
en h definition,) par ordrc rous cnfcmble c<3mc s'cnfuic.
Lcscbiraderesligmrians qu3mitcz,dcfquel$ 1'cxpli-
cation fc trouvc cs 14.15.16 ^7.18. definitions, font tcls.
©Coramcnctmcnt dc quantusi oui eft nombro Atith.
. pu radical quclconquc..
0 prime quamite.
(2, lecondcquanurc.
(<} tierce quamitc.
0 quartc qiiamirc ,ix c.
Lcs ch.ua ctetc> iignili.uis poOpofccs qunnrircz,
dcfquels 1'cxplication fc trouvc d Li 18 definition,
font tcls:
1 icc,»j Vnc prime quandti (ccondcmcncpofcc.
j) C^unttc Tccondcs qiuntitcz tict'cemciu pofcci,
ou pioccdans de la prime quantitc dctcc-
mcnt pofcc.
i (|j Tcc0 Protluicl d'une prime quantitc par unc prime
'qu.inrirc fccondeincntpofcc.
5 0 tcr(7}Piodui£k de cincq quattesquantitez par itnc
rccondcquintiteticrccmtnt pofcc.
Lcs charadlcics n^nifians ndnc dc'quels {'expli-
cation ft1 trouvc a la ip (k 30 clctimuoa iont tcls :
4/ • Ratine dc qu.irrc.
Hi/ Racine de racine dc quarre. '
t*4/ •-' Racine dc racine dc racine dc quarre.
#uv^ R.ictnc (ic racine dc racine de racine dc quarre,
4/C<) Racine dc cube.
<*/ (?) Racuic*de racine dc cube,
4/ (j)R;icinc dcquirrfe quantitc.
4</vi)Raciocdc racine dc quane cjuantitc,&.'c.
Lc characlcic /Iguifiant b feparation entic le fi-
gne de racine & la quautitc, duqucl 1'cxplication ic
trouvc i la 34. definition, cfbtcl. ^
X» Comme i/ 3 X® n'c't p«ls le mcfinc q[ue </ 5 (71,
cominc did eft a lididtc 54. definition.
Les chaia&cics Jlgnifiaiis plus <Sc moiiw, comme 1
la $6 dctmition, font tcls :
— Moins.
Et pour cxpliquer la racine d'un multlnomic
Cqu'aucuns arpcllcm vacinc univcricllc) nousule-
rons le vocable da multinomic, comme:
4/bino i,ri~'<v $• c'eft a dire racine quarrccdc bino-
mie,oudc laiommcdc z & 4/5.
«/ tiino V ,3,-r V i — V 5 , c'cft a dire Vacinc quarrc'c
dc rrinomie , ou de la iommc de */ 3 &c */ i <?c —
!X'i^i-/i/ •, c 'eft a dire racine cubique de
c'cftadifc racine quarrcc dcbino- j
*/©fein^
binonne
/l/bino ;
f ® bino i Qf; -j- 1 CD> c'eft a dire racine cubique dc bi«
notnic i (1^, -f- 1 0, &c.
FIG. 69. — From S. Stevin;s L'arithmttique
in (Euvres math&matiques (ed. A. Girard;
Leyden, 1634), p. 19.
160 A HISTORY OF MATHEMATICAL NOTATIONS
The book contains other notations which are not specially ex-
plained. Thus the cube of B+C is given in the form B(Bq+C3Q) +
C(Bl+Cq).
We see here the use of round parentheses, which we encountered
before in the Algebra of Clavius and, once, in Cardan. Notice also
that C\ means here 3C2.
Autre exemple In Modern Symbols
"Soit 1(3) esgale & -G(l)+20 Let x» = -6z+20
Divisons tout par 1(1) Divide all by x,
20 90
1(2) esgale a -6+ ." x2= -6+ .
Again (fol. F3) : "Soit 1(3) esgale a 12(1) - 18 (impossible d'estre esgal)
car le ^ est 4 9 qui est \ do 18
son cube 64 81 son quarr£ .
Et puis que 81 est plus que 64, Pequation est im-
possible & inepte."
Translation: "Let 0?= 12z— 18 (impossible to be equal)
because the i is 4 9 which is \ of 18
its cube 64 81 its square
And since 81 is more than 64, the equation is im-
possible and inept. "
A few times Girard uses parentheses also to indicate multiplica-
tion (see op. tit., folios Cf, D?, F§.
GERMAN-SPANISH: MARCO AUREL
(1552)
165. Aurel states that his book is the first algebra published in
Spain. He was a German, as appears from the title-page: Libra
primer o de Arithmetica Algebratica ... por Marco Aurel, natural Aleman
(Valencia, 1552). l It is due to his German training that German alge-
braic symbols appear in this text published in Spain. There is hardly
a trace in it of Italian symbolism. As seen in Figure 70, the plus (+)
and minus (— ) signs are used, also the German symbols for powers of
the unknown, and the clumsy Rudolffian symbols for roots of different
1 Aurel's algebra is briefly described by Julio Rey Pastor, Los malhemdlicos
espaftoksdelsigloXVI (Oviedo, 1913), p. 36 n.; see Bibliotheca mathematica, Vol.
IV (2d ser., 1890), p. 34.
INDIVIDUAL WRITERS
161
orders. In place of the dot, used by Rudolff and Stifel, to express the
root of a polynomial, Aurel employs the letter v, signifying universal
root or rayz vniuersal. This v is found in Italian texts.
Fio. 70. — From Aurel's Arithmctica algebratica (1552). (Courtesy of the Li-
brary of the University of Michigan.) Above is part of fol. 43, showing the -f and
— , and the radical signs of Rudolff, also the y'v. Below is a part of fol. 73B, con-
taining the German signs for the powers of the unknown and the sign for a given
number.
162 A HISTORY OF MATHEMATICAL NOTATIONS
PORTUGUESE-SPANISH: PEDRO NUNEZ
(1567)
166. Nunez' Libra de algebra (1567)1 bears in the Dedication the
date December 1, 1564. The manuscript was first prepared in the
Portuguese language some thirty years previous to Nunez' prepara-
tion of this Spanish translation. The author draws entirely from
Italian authors. He mentions Pacioli, Tartaglia, and Cardan.
The notation used by Nunez is that of Pacioli and Tartaglia. He
uses the terms Numero, cosa, censo, cubo, censo de censo, relato primo,
censo de cubo or cubo de censo, relato segundo, censo de censo de ceso,
cubo de cubo, censo de relato primo, and their respective abbreviations
co., ce., cu., ce.ce., re.p°, ce.cu. or cu.ce., re.seg0. ce.ce.ce., cu.cu., ce.re.p0.
He uses p for mas ("more"), and m for menos ("les"). The only use
made of the •%* is in cross-multiplication, as shown in the following
, /f i At\ tt A.- i 12. 2.cu.p.8.
sentence (fol. 41): ... partiremos luego y — por — y — como si
jL.cOm ji.ce,
fuessen puros quebrados, multiplicado en *|«, y verna por quociente
12 ce
rt '— g el qual quebrado abreuiado por numero y por dignidad
verna a este quebrado T~~"''>'J~'" This expression, multiplicando en
*f«, occurs often.
Square root is indicated by R., cube root by R.cu., fourth root by
R.R., eighth root by R.R.R. (fol. 207). Following Cardan, Nunez
uses L.R. and R.V. to indicate, respectively, the ligatura ("combina-
tion") of roots and the Raiz vniuersal ("universal root," i.e., root of a
binomial or polynomial). This is explained in the following passage
(fol. 456): "... diziendo assi: L.R.7pRA.p.3. que significa vna quanti-
dad sorda compuesta de .3. y 2. que son 5. con la R.7. o diziendo assi:
L.R.3p2.co. Raiz vniuersal es raiz de raiz ligada con numero o con
otra raiz o dignidad. Como si dixessemos assi: R.v. 22 p 7?9."
Singular notations are 2. co. \. for 2[z (fol. 32), and 2. co. | for
2fz (fol. 366). Observe also that integers occurring in the running
text are usually placed between dots, in the same way as was custom-
ary in manuscripts.
Although at this time our exponential notation was not yet in-
vented and adopted, the notion of exponents of powers was quite well
understood, as well as the addition of exponents to form the product
1 Libra de Algebra en arithmetica y Geometria. Compuesto por el Doctor Pedro
Nunez, Cosmographo Mayor del Rey de Portugal, y Cathedratico Jubilado en la
Cathedra de Mathematicas en la VniuerMad de Coymbra (En Anvers, 1567).
INDIVIDUAL WRITERS 163
of terms having the same base. To show this we quote from Nunez
the following (fol. 266) :
"... si queremos multiplicar .4. co. por .5. ce. dircrnos asi .4. por
.5. hazen .20. y porque .1. denominacio de co. sumado con .2. de-
nominacion de censo hazen .3. quc cs denominacio de cubo. Diremos
por tanto q .4. co. por .5. ce. hazen .20. cu. ... si multiplicamos .4.
cu. por .8. ce.ce. diremos assi, la denominacion del cubo es .3. y la
denominaciS del censo de censo es .4. q sumadas haze .7. q sera la
denominacio dela dignidad engedrada, y por que .4. por .8. hazen .32.
diremos por tanto, que .4. cu. multiplicados por .8. ce.ce. hazen .32.
dignida-dcs, que tienen .7. por denominacion, a quc Hainan relatos
segundos."
Nunez' division1 of 12z3+18x2+27a;+17 by 4z+3, yielding the
I1 2
quotient 3x2+2Jx+51V+^ VQ > *s as
~-
"Partidor A.co.p.3 I I2.cu.p.l8.ce.p.27.co.p.l7.
12.cu.p. 9.ce.
9. ce. p. 27. co. p. 17.
O.ce.p. G.co.J.
20.co.-Jr.j5. 15
par A.co.p.3."
Observe the "20.co.y for 20|x, the symbol for the unknown appearing
between the integer and the fraction. _______ ___
Cardan's solution of x3+3x = 36 is ^1/325+ 18 -^1/325 -18,
and is written by Nunez as follows:
R. V.cu.R.325.p.l8.m.R. V .cu. .fl.325. w. 18.
As in many other writers the V signifies vniversal and denotes, not
the cube root of 1/325 alone, but of the binomial 1/325+18; in other
words, the V takes the place of a parenthesis.
1 See II. Bosnians, "Sur le 'Libro de algebra' de Pedro Nunez," Bibliotheca
mathematica, Vol. VIII (3d ser., 1908), p. 160-62; see also Tropfke, op. tit. (2d ed.),
Vol. Ill, p. 136, 137.
164 A HISTORY OF MATHEMATICAL NOTATIONS
ENGLISH: ROBERT RECORDE
(i ^/i^r?! 1 f^7\
\lcr±O[IJ, JuJOl )
167. Robert Recorders arithmetic, the Grovnd of Aries, appeared
in many editions. We indicate Recorders singular notation for pro-
portion:1
(direct) 3:8 = 16s.: 42s. 8d.
2s. 8d.
Z1
(reverse)
There is nothing in Recorders notation to distinguish between the
"rule of proportion direct" and the "rule of porportion reverse/' The
difference appears in the interpretation. In the foregoing "direct"
proportion, you multiply 8 and 16, and divide the product by 3. In
the "reverse" proportion, the processes of multiplication and division
are interchanged. In the former case we have 8X16-^3 = x, in the
second case we have i-X-rV^iV^^- I*1 both cases the large strokes in
2 serve as guides to the proper sequence of the numbers.
168. In Recorders algebra, The Whetstone of Witte (London,
1557), the most original and historically important is the sign of
equality ( = ), shown in Figure 71. Notice also the plus (+) and minus
( — ) signs which make here their first appearance in an English book.
In the designation of powers Recorde uses the symbols of Stifel
and gives a table of powers occupying a page and ending with the
eightieth power. The seventh power is denoted by 6j"$; for the
eleventh, thirteenth, seventeenth powers, he writes in place of the
letter b the letters c, d, E, respectively. The eightieth power is de-
noted by SJjgJs, showing that the Hindu multiplicative method of
combining the symbols was followed.
Figure 72 shows addition of fractions. The fractions to be added
are separated by the word "to." Horizontal lines are drawn above
and below the two fractions; above the upper line is written the new
numerator and below the lower line is written the new denominator.
In "Another Example of Addition," there are added the fractions
5s6+3s5 , 20s3 -6s5
~ r and yr~7i " •
Qx9 bx*
I0p. tit. (London, 1646), p. 175, 315. There was an edition in 1543 which was
probably the first.
INDIVIDUAL WRITERS 165
Square root Recorde indicates by j/. or j/$, cube root by
V\A/. or /vw/.cC- Following Rudolff, he indicates the fourth root by
Thejfrte
aa tfictc foojfef s Doe crtentte ) to &f ftinetc it ottclp fnf o
ttooo parted Cfflftereof the firfte is, ife£r» <w* nombcris
cqutlle wto one other. <3tlD tfte fccotlDc (0 >>6oi ow neat;
irr is compared as ejtullc »nt9,wtbcrn9mbers,
fllluaics toillgngpou to rcmrbcr, tljatpou reduce
sournombew, totficirleaftc Denominations, aim
fmalltfte fo;mc^bcfo^e pott pjoce&c anp farther*
flno again,if pour <!*** ion be focftc, tftat tfte grea^
tttte ucnomtnation G/?^ be toincD to anp partc of 4
compounDc nombcr 9 ?ou ftall tottrne It To , tfjat the
tiombcroftbcgrcatcttc Co^ne alone, maicffanoca^
cquallctotficrcfte*
ilnD tbts ts all tbat ncaoctb to be taugljtc , conccr-
,fo^ caffc altcratto of cftMtiom.% tuill p?0'
pounoc a fctoc crapl£5,b(caufe tf>c extraction of tfjetr
rootc0,matc tt>c mo?c aptlp bee iu;ottgl)te» flnD to a*
uoiDctbctctrioufe repetition of tftcfe U)00^cs:t3c-
qualle to : 3 tuill fette asj uoe often in tooo&efcfe,*
paire of parallels, o: <Dcmotoc lines of one lengtbe,
tl)us:^-===,btcaufe noe.2,
cquallc.
*I J.f «— -7 I*
20,t£. -
26.5* — I —
6. $45. -- 12^—40^— I— 480^ — 9.5-
• jn tbc firfte there appeared 2 » nombcra , that ts
I4.^/
FIG. 71.— From Robert Recorded Whetstone of Witte (1557)
-J but Recorde writes it also /vv/^- Instructive is the dialogue on
these signs, carried on between master and scholar:
166 A HISTORY OF MATHEMATICAL NOTATIONS
"Scholar: It were againste reason, to take reason for those signes,
whiche be set voluntarily to signifie any thyng; although some tymes
there bee a certaine apte conformitic in sochc thyngcs. And in these
Jti other Example \)f Addition.
fertcrme0*
noe multiplication, no? reouttfon to one
common DenomtnatojtftI) tbef bee one all reat»p:no<
tber ran the nombers be rcDuceD, to an? otber Icffcr^
but tbe quantities onel? be re&ticeD as pou fee*
Scholar. 3|p;atet?ouletmep;oue»
jfn otberExamflf.
— \— 9
°
oaarttc pour iuo;fec tuell, before pou re*
ourett.
^cbolar, 31 Teem? faulted ftauc frttc.2. nombcra
fcucrallp, luttb one figne G/5'4^; : bp reafon 31 DID not
fojcf(c,tl)at,ct.multipJicD U)tti;,<£.0oct^ maUe the
tiUc
FIG. 72.— Fractions in Recorders Whetstone of Witte (1557)
figures, the nomber of their minomes, seameth disagreable to their
order.
"Master: In that there is some reason to bee thewed: for as .j/.
declareth the multiplication of a nomber, ones by it self; so ./wV-
representeth that multiplication Cubike, in whiche the roote is repre-
INDIVIDUAL WRITERS
167
sen ted thrise. And ./vsA standeth for .j/./\/. that is .2. figures of
Square multiplication: and is not expressed with .4. minomes. For
so should it seme to expresse moare then .2. Square multiplications.
But voluntarie signes, it is inoughe to knowe that this thei doe signifie.
fUF;
iW»/f< :.. J*' : '*
FJG. 73.— Radicals in Recorded Whetstone of Witte (1557)
And if any manne can diuise other, moare easie or apter in use, that
maie well be received."
Figure 73 shows the multiplication of radicals. The first two
exercises are fOlXI^ 12 = 1^092, ^7|x^| = ^&t. Under fourth
roots one finds V 15X^7== I^IOS .
168 A HISTORY OF MATHEMATICAL NOTATIONS
ENGLISH: JOHN DEE
(1570)
169. John Dee wrote a Preface to Henry Billingsley's edition of
Euclid (London, 1570). This Preface is a discussion of the mathe-
matical sciences. The radical symbols shown in Figure 74 are those
of Stifel. German influences predominated.
I^<5Hfet)^
nfibers,
^yS^pre & Ldfcas tlius </ft ii '+ Vc€. if.Or thusV^S* i$>
. &c.Arid&me tytne ivfth'ivhble numbers, or faftions of whole
FIG. 74. — Radicals, John Dee's Preface to Billingsley's edition of Euclid
(1570).
In Figure 75 Dee explains that if a:b = c:d, then also a:a — b —
c:c — d. He illustrates this numerically by taking 9:6 = 12:8. Notice
Dee's use of the word "proportion" in the sense of "ratio." Attention
is drawn to the mode of writing the two proportions 9.0:12.8 and
9.3:12.4, near the margin. Except for the use of a single colon (:),
[ conucrfion ofproj^ttio^andoffomccucrfipnofproporrionJLA
, berSjasp.to^fow.toS.c^
the cacccfTc of ^.thc antecedent of the firft proportion abouc 9.6: n . 8
^thcconfcqaedtofdieramcis3: the txcdftpfii; the ante- o ,a ; w 0 4,
ccdcnt ofthcfecond proportion abouc 8,thec6ftfeq«icnt of J ' * *
xhcfime^4:nowc6pai^thcameccdeiuoftheftilpro
nonp.as^tccedcttoj.thcexcciretherofaboue^v&cconfcqu^
; fequent , Kkcwife compare i«. thcahteccdcntoftheftcond proportion as antece-
j detit to ^thc«ceffetherofalx)uc 8, the confequcnt^ to his con^
| yournumbersbein oysoidcr by conuerfion of proportion: as p,to 3 :fo 11^64:
FIG. 75. — Proportion in John Dee's Preface to Billingsley's edition of Euclid
(1570).
in place of the double colon ( : : ) , this is exactly the notation later used
by Oughtred in his Clavis mathematicae. It is possible that Oughtred
took the symbols from Dee. Dee's Preface also indicates the origin of
these symbols. They are simply the rhetorical marks used in the text.
See more particularly the second to the last line, "as 9. to 3: so 12.
to 4:"
INDIVIDUAL WRITERS 169
ENGLISH: LEONARD AND THOMAS DIGGES
(1579)
170. The Stratioticos1 was brought out by Thomas Digges, the
son of Leonard Digges. It seems that the original draft of the book
was the work of Leonard; the enlargement of the manuscript and its
preparation for print were due to Thomas.
The notation employed for powers is indicated by the following
quotations (p. 33) :
"In this Arte of Numbers Cossical, wae proceede from the Roote
by Multiplication, to create all Squares, Cubes, Zenzizenzike, and
Stir Solides, wyth all other that in this Science are used, the whyche
by Example maye best bee explaned.
12345678 9 10 11 12
Roo. Sq. Cu. SqS. Sfo. SqC. Bfs. SSSq. CC. Sfs. CfS. 8SC. "
2 4 8 10 32 04 128 256 512 1024 2048 4090
Again (p. 32) :
". ... Of these [Roote, Square, Cube] are all the rest com-
posed. For the Square being four, againe squared, maketh his
Squared square 16, with his Character oner him. The nexte being not
made by the Square or Cubike, Multiplication of any of the former,
can not take his name from Square or Cube, and is therefore called a
Surd solide, and is onely created by Multiplicand of 2 the Roote, in
16 the SqS. making 32 with his couenient Character ouer him & for
distinctio is tearmed y first Surd solide .... the nexte being 128, is
not made of square or Cubique Multiplication of any, but only by the
Multiplication of the Squared Cube in his Roote, and therefore is
tearmed the B.S.solide, or seconde S. solide
'This I have rather added for custome sake, bycause in all parts
of the world these Characters and names of Sq. and Cu. etc. are used,
but bycause I find another kinde of Character by my Father demised,
farre more readie in Multiplications, Diuisions, and other Cossical
operations, I will not doubt, hauing Reason on my side, to dissent
from common custome in this poynt, and vse these Characters en-
suing: [What follows is on page 35 and is reproduced here in Fig. 76]."
1 An Arithmeticall Militare Treatise, named Stratioticos: compendiously teaching
the Science of Nubers, as well in Fractions as Integers, and so much of the Rules and
A equations Algebraicall and Arte of Numbers Cossicall, as are requisite for the Profes-
sion of a Soldiour. Together ivith the Moderne Militare Discipline, Offices, Lawes and
Dueties in euery wel gouerned Campe and Armie to be observed: Long since attepted
by Leonard Digges Gentleman, Augmented, digested, and lately finished, by Thomas
Digges, his Sonne .... (At London, 1579).
170
A HISTORY OF MATHEMATICAL NOTATIONS
FIG. 76. — Leonard and Thomas Digges, Stratioticos (1579), p. 35, showing the
unknown and its powers to x9.
INDIVIDUAL WRITERS 171
As stated by the authors, the symbols are simply the numerals
somewhat disfigured and crossed out by an extra stroke, to prevent
confusion with the ordinary figures. The example at the bottom of
page 35 is the addition of 20x+3Qx*+25x? and 45z+16o;2+13r'. It
is noteworthy that in 1610 Cataldi in Italy devised a similar scheme for
representing the powers of an unknown (§ 340).
The treatment of equations is shown on page 46, which is re-
produced in Figure 77. Observe the symbol for zero in lines 4 and 7;
this form is used only when the zero stands by itself.
A little later, on page 51, the authors, without explanation, begin
to use a sign of equality. Previously the state of equality had been
expressed in words, "equall to," "are." The sign of equality looks as
if it were made up of two letters C in these positions OC and crossed
by two horizontal lines. See Figure 78.
This sign of equality is more elaborate than that previously de-
vised by Robert Recorde. The Digges sign requires four strokes of
the pen; the Recorde sign demands only two, yet is perfectly clear.
The Digges symbol appears again on five or more later pages of the
Stratioticos. Perhaps the sign is the astronomical symbol for Pisces
("the Fishes"), with an extra horizontal line. The top equation on
page 51 isz2
ENGLISH: THOMAS MASTERSON
(1592)
171. The domination of German symbols over English authors of
the sixteenth century is shown further by the Arithmdicke of Thomas
Masterson (London, 1592). Stifel's symbols for powers are used. We
reproduce (in Fig. 79) a page showing the symbols for radicals.
FRENCH: JACQUES PELETIER
(1554)
172. Jacques Peletier du Mans resided in Paris, Bordeaux, Be-
ziers, Lyon, and Rome. He died in Paris. His algebra, De occvlta Parte
Nvmerorvm, Quam Algebram vocant, Libri duo (Paris, 1554, and several
other editions),1 shows in the symbolism used both German and
Italian influences: German in the designation of powers and roots,
done in the manner of Stifel; Italian in the use of p. and m. for "plus"
and "minus."
1 All our information is drawn from H. Bosnians, "L'algebre de Jacques
Peletier du Mans," Extrait de la revue des questions scientifiques (Bruxelles: Janu-
ary, 1907), p. 1-61.
172
A HISTORY OF MATHEMATICAL NOTATIONS
on) to $> oa t&e«wiNt#, t&ere maotiw fome fflttn&rr (a
• -^omctimf* Rcdmfbon fct maUebpuwjina foitfth?r all '
wwjis^^
'"^J' " ^4*' ~ 'J ti" ~ **** "Or %
l
' «i*r^rti'*ti4i^i&]^
A. HL+ I'.^K'JUtttra.'S&i^ iiM':''«*-*i— ati.i^iii, ^.'liJr •f!*Tr**Tf •
4
t«».'
^•gi«i*ii««
Fia. 77. — Equations in Digges, Stratioticos (1579)
INDIVIDUAL WRITERS
173
— ¥ J
FIG. 78. — Sign of equality in Digges, Straiioticos (1579). This page exhibits
also the solution of quadratic equations.
AMTHMBTICKB. LIB.
75 "*• »
80
9 f ^50 —
/4f/f i 4
48
27
8 1 +
oi
FIG. 79.— Thomas Masterson, Arithmeticke (1592), part of p. 45
174 A HISTORY OF MATHEMATICAL NOTATIONS
Page 8 (reproduced in Fig. 80) is in translation: "[The arith-
metical progression, according to the natural order of counting,]
furnishes us successive terms for showing the Radicand numbers
and their signs, as you see from the table given here [here appears the
table given in Fig. 80].
: ''T';^
V
•
r ^U'^v1;1, ~'i
~ ', ' ,.,/-,*•- . _ . , - i - ' , j < ' ^M"-i!"*;s.'V ' '- *
" ; ../' ;U < r " " ' ; M '• /^yl*.V-:"'-'J.'t
;',;OiiJr */V'4 1* :?» -7»
.-.I, K, |, I/I, ,*
.-; ,f, iy 4, S, 'itf, lit, jfcjtf, f^
', : j>i i'^t,*-' /ij, ^14 • if,' V^c^ 'T. ,;
.,:
;-l':^':l;1:'>,r-" - ' / , "; v,^:-, M
' ; ^ , ! $t & ! |
i ' E qrii ft 5H i;fe|
4c IP ^ r |f ijy ^
; " 4c c^ 1^
;:;(!
."
'
;.
;
VI!
FIG. 80, — Designation of powers in J. Peletier's Algebra (1554)
"In the first line is the arithmetical progression, according to the
natural order of the numbers; and the one which is above the &
numbers the exponent of this sign & ; the 2 which is above the 3 is the
exponent of this sign 3 ; and 3 is the exponent of c, 4 of 33, and so on.
"In the second line are the characters of the Radicand numbers
INDIVIDUAL WRITERS
175
which pertain to algebra, marking their denomination." Then are ex-
plained the names of the symbols, as given in French, viz., 1} ratine,
3 Qanse, <£ cube, etc.
FIG. 81. — Algebraic operations in Peletier's Alytbra (1554)
Page 33 (shown in Fig. 81) begins with the extraction of a square
root and a "proof" of the correctness of the work. The root extraction
is, in modern symbols:
36z4+48z3 - 104x2 - 80x+ 100
+ 12x2+ 8z-
+120z2+80z-100.
176 A HISTORY OF MATHEMATICAL NOTATIONS
The "proof" is thus:
4x~ 10
6z2+ 4x - 10
36z4+24z3- 60s2
16s2-'
- 60z2-40z+100
Further on in this book Peletier gives:
1/3 15 p. |/j8, signifying 1/15+1/8 .
l/j . 15 p. i/j8, signifying ^15+1/8 .
FRENCH: JEAN BXJTEON
(1559)
173. Deeply influenced by geometrical considerations was Jean
Buteon,1 in his Logistica quae et Arithmetica vulgo dititur (Lugduni,
1559). In the part of the book on algebra he rejects the words res,
census, etc., and introduces in their place the Latin words for "line,"
"square," "cube," using the symbols p, <>, Q). He employs also P and
My both as signs of operation and of quality. Calling the sides of an
equation continens and contentum, respectively, he writes between
them the sign [ as long as the equation is not reduced to the simplest
form and the contentumt therefore, not in its final form. Later the
contentum is inclosed in the completed rectangle [ ]. Thus Buteon
writes 3p M 7 [ 8 and then draws the inferences, 3p [15], lp [5]. Again
he writes | <> [100, hence 1<> [400], lp [20]. In modern symbols:
3z-7 = 8, 3z = 15, re = 5; iz2<=100, ^ = 400, x = 20. Another example:
1 Q P 2 [218, i a [216, 1 O [1728], lp [12]; in modern form la;3+2 =
218, i*3 = 216, z* = 1,728, x = 12.
When more than one unknown quantity arises, they are repre-
sented by the capitals A, By C. Buteon gives examples involving only
positive terms and then omits the P. In finding three numbers sub-
ject to the conditions x+^y+^z** 17, y+&+ 1*= 17, z+-±x+±y = 17,
he writes:
IA , \B , |C [17
IB , tA , iC [17
1C , \A , \B [17
1 Our information is drawn from G. Werthheim's article on Buteon, Biblio-
theca mathematica, Vol. II (3d ser., 1901), p. 213-19.
INDIVIDUAL WRITERS
and derives from them the next equations in the solution:
2A . IB . 1C [34
1A . 35 . 1C [51
1A . IB . 4C [68, etc.
177
$m."®aM*\i^i4ffi7* *¥**_
"it 5 ,fiti»rfjidito 4,£»f)
FIG. 82.— From J. Buteon, Arilhmetica (1559)
178 A HISTORY OF MATHEMATICAL NOTATIONS
In Figure 82 the equations are as follows:
34 + 125+ 3(7= 96
3A+ 1B+ 1C- 42
2(7= 54
3A+ 35+ 15(7 = 120
3A+ 15+ 1(7 = 42
25+ 14C = 78
225+154(7 = 858
225+ 4(7=108
150(7 = 750
FRENCH: GUILLAUME GOSSELIN
(1577)
174. A brief but very good elementary exposition of algebra was
given by G. Gosselin in his De arle magna, published in Paris in 1577.
Although the plus (+) and minus (— ) signs must have been more or
less familiar to Frenchmen through the Algebra of Scheubel, published
in Paris in 1551 and 1552, nevertheless Gosselin does not use them.
Like Peletier, Gosselin follows the Italians on this point, only Gosselin
uses the capital letters P and M for "plus" and "minus," instead of
the usual and more convenient small letters.1 He defines his notation
for powers by the following statement (chap, vi, fol. v) :
L - 2 - Q • 4 • C - 8 • QQ • 16 • RP • 32 • QC • 64 - RS • 128 - CC • 512 .
Here RP and RS signify, respectively, relatum primum and relatum
secundum.
Accordingly,
11 12L M IQ P 48 aequalia 144 M 24L P 2Q "
means
*Our information is drawn mainly from H. Bosnians' article on Gosselin,
Bibliotheca mathematica, Vol. VII (190(>-7), p. 44-66.
INDIVIDUAL WRITERS
179
The translation of Figure 83 is as follows:
" . . . . Thus I multiply 4z-6z2+7 by 3x2 and there results
12Z3— 18z4+21a;2 which I write below the straight line; then I multi-
i }',' H' 'i' '' '
1 ' ,
Ij ti»-w^i|> » r ' jT if ^ i f « Mt IK Ji r* f '.iff f ^Sr " sM) "'"" T f1 1;, ITV ^~i jti
^>'y& j$.ihkxGt&m&' -
FIG. 83.— Fol. 45t>° of Gosselin's Z>e arie magna (1577)
ply the same 4z— 6o;2+7 by +4x, and there results +16x2— 24^
+28z; lastly I multiply by —5 and there results — 20x+30x2— 35.
180 A HISTORY OF MATHEMATICAL NOTATIONS
And the sum of these three products is 67x*+8x— 12s3— 18s4— 35, as
will be seen in the example.
4z - 6z2+ 7
3z2+ 4x - 5
( 12z3-18o;4+21z2
Products \ 16z2- 24z3+28z
[-20s+30z2-35
Sum 67z2+8z-12z3-18z4-35 .
On the Division of Integers, chapter viii
Four Rules
+ divided in + the quotient is +
— divided in — the quotient is +
— divided in + the quotient is —
+ divided in — the quotient is — "
175. Proceeding to radicals we quote (fol. 475): "Est autem
laterum duplex genus simplicium et compositorum. Simplicia sunt
19, LC8, LL16, etc. Composita vero ut LF24 P L29, LF6 P L8."
In translation: "There are moreover two kinds of radicals, simple
and composite. The simple are like 1/9, f^8, 1/16, etc. The com-
posite are like */24+)/29, ^G+l/S." First to be noticed is the dif-
ference between L9 and 9L. They mean, respectively, 1/9 and Qx. We
have encountered somewhat similar conventions in Pacioli, with whom
& meant a power when used in the form, say, "5 . 5?" (i.e., x4), while
B, meant a root when followed by a number, as in # .200. (i.e., 1/200)
(see § 135). Somewhat later the same principle of relative position
occurs in Albert Girard, but with a different symbol, the circle.
Gosselin's LV meant of course latus universale. Other examples of his
notation of radicals are L7L10 P L5, for */VlO+ 1/5, and LFCL5
PLC10 for
In the solution of simultaneous equations involving only positive
terms Gosselin uses as the unknowns the capital letters A, B, C, . . . .
(similar to the notation of Stifel and Buteon), and omits the sign
P for "plus"; he does this in five problems involving positive terms,
following here an idea of Buteo. In the problem 5, taken from Buteo,
Gosselin finds four numbers, of which the first, together with half of
the remaining, gives 17; the second with the third of the remaining
gives 12; and the third with a fourth of the remaining gives 13; and
INDIVIDUAL WRITERS 181
the fourth with a sixth of the remaining gives 13. Gosselin lets A, Bt
C, D be the four numbers and then writes:
Modern Notation
lia 17 , x+^y+{z+^w = l7 ,
lia 12, etc. " y+fc+fc+b™ = 12 .
He is able to effect the solution without introducing negative terms.
In another place Gosselin follows Italian and German writers in
representing a second unknown quantity by q, the contraction of
quantitas. He writes (fols. 84#, 85A) "1L P 2q M 20 aequalia sunt
1L P 30" (i.e., lz+22/-20=lz+30) and obtains "2q aequales 50, fit
1^25" (i.e., 2y = 50, i/ = 25).
FRENCH: FRANCIS VIETA
(1591 and Later)
176. Sometimes, Vieta's notation as it appears in his early publi-
cations is somewhat different from that in his collected works, edited
, „ a , , . __ „ , , 3£Z)2-3#A2
by Fr. van Schooten in 1646. For example, our modern - ^ -
is printed in Vieta's Zeteticorum libri v (Tours, 1593) as
" B in D quadratum 3 — B in A quadratum 3 "
_ f
while in 1646 it is reprinted1 in the form
"gin DqZ-Bm A? 3 >f
4
Further differences in notation are pointed out by J. Tropfke:2
Zeteticorum libri v (1593)
f B in A 1
^ . OT> " B in A , I -B in H I , , D „
Fol. 3B: — TT -- 1- s - pj - > aequabuntur B ."
V I * )
,T , Bx , Bx-B • H D
Modern : -_- -\ -- ^ - = B .
LJ r
OK K
Lib. II, 22: «i££_i?.»
o o
1 Francisci Victae Opera mathemalica (ed. Fr. & Schooten; Lvgdvni Batavorvm,
1646), p. 60. This difference in notation has been pointed out by H. Bosnians, in
an article on Oughtred, in Extrait des annales de la soci&& scientifique de Bruxelles,
Vol. XXXV, fasc. 1 (2d part), p. 22.
2 Op. cit., Vol. Ill (2d ed., 1922), p. 139.
182 A HISTORY OP MATHEMATICAL NOTATIONS
Modern: B(D*+BD) .
Lib. IV, 20: - D in (^cubum2l „
' i — D cubo J
Modern: D(2B3-D*) .
Van Schooten edition of Vieta (1646)
^ .„ " B in A , B in A-B in # ... D „
P. 46: — ^ -- 1 -- ET— - aequabitur B ."
D r
/25 /5 „
\y \3-
P. 70: " B in D quad.+# in D ."
P. 74: " D in 5 cubum 2-D cubo ."
Figure 84 exhibits defective typographical work. As in StifePs
Arithmetica Integra, so here, the fractional line is drawn too short.
In the translation of this passage we put the sign of multiplication
(X) in place of the word in: ". . . . Because what multiplica-
tion brings about above, the same is undone by division, as
^ . i.e.. jj. \ and ^ is A. *
o r>
... A2 A2-\-ZXB
Thus in additions, required, to -~ to add Z. The sum is - ^ - ;
. . „ A1 . , , Z2 — .
or required, to -^ to add 77 . The sum is
£> Or
A2
In subtraction, required, from -^- to subtract Z. The remainder is
n . . . A2 , u, . Z2 _ . , .
0 Qr required, from -^ to subtract -~ . Ine remainder is
£> Cr
Observe that Vieta uses the signs plus (+) and minus (—), which
had appeared at Paris in the Algebra of Scheubel (1551). Outstanding
in the foregoing illustrations from Vieta is the appearance of capital
letters as the representatives of general magnitudes. Vieta was the
first to do this systematically. Sometimes, Regiomontamus, Rudolff,
Adam Riese, and Stifel in Germany, and Cardan in Italy, used letters
at an earlier date, but Vieta extended this idea and first made it an
INDIVIDUAL WRITERS A 183
essential part of algebra. Vieta's words,1 as found in his Isagoge, are:
"That this work may be aided by a certain artifice, given magnitudes
are to be distinguished from the uncertain required ones by a symbol-
ism, uniform and always readily seen, as is possible by designating the
required quantities by letter A or by other vowel letters Ayly0y V, Y,
and the given ones by the letters #, (?, D or by other consonants,"2
Vieta's use of letters representing known magnitudes as coeffi-
cients of letters representing unknown magnitudes is altogether new.
In discussing Vieta's designation of unknown quantities by vowels,
[Tluper cttecit multiplicatio , idem rcioiuit diuilid vf'B W~ A~714i» 4% "ff :g W
I eft A planum , , > ' ~H§ » ' V '""IP*
' ** ! < • t t ? - '• v>*> **•(? lJ ' l&l' t7 1
* ' ' ' x
• Woe in AddfcioBibw* , Oportei* A jftyw fdifcr* Vj^^w jafc? ^ffiW- ' • •<!' f I i
" JV * A plano-i^duccrc ,^ qm4rtQiro. ; Rcfiduaerit tAipittt«»,ite"G ^ t!*
t~~~ =m^^ ^ t .. • * <§»>»•» x^' » ,'».>, M •,.Jk.j-A^ i.. t> } ,*
FIG. 84. — From Vieta's In arlem analyticam Isagoge (1591). (I am indebted to
Professor H. Bosnians for this photograph.)
C. Henry remarks: "Thus in a century which numbers fewer Oriental-
ists of eminence than the century of Viet a, it may be difficult not to
regard this choice as an indication of a renaissance of Semitic lan-
guages; every one knows that in Hebrew and in Arabic only the conso-
nants are given and that the vowels must be recovered from them/'3
177. Vieta uses = for the expression of arithmetical difference.
He says: "However when it is not stated which magnitude is the
greater and which is the less, yet the subtraction must be carried out,
1 Vieta, Opera mathematica (1646), p. 8.
2 "Quod opus, ut arte aliqua juvetur, symbolo constant! et perpetuo ac bene
6onspicuo date magnitudines ab incertis quaesititiis distinguantur, ut pote magni-
tudines quaesititias elemento A aliave litera vocali, E, 7, O, V, Y datas elementis
Bj G, Dt altisve consonis designando."
3 "Sur Forigine de quelques notations mathe'matiques," Revue archfologique,
Vol. XXXVIII (N.S., 1879), p. 8.
184 A HISTORY OF MATHEMATICAL NOTATIONS
the sign of difference is =, i.e., an uncertain minus. Thus, given A2
and B2, the difference is A2—B2, or £2=AV'1
We illustrate Vieta's mode of writing equations in his Isagoge:
"B in A quadratum plus D piano in A aequari Z solido," i.e., BA2+
D2A = Z3, where A is the unknown quantity and the consonants are
the known magnitudes. In Vieta's Ad Logisticen speciosam notae
priores one finds: "A cubus, +A quadrato in B ter, +A in B
quadratum ter, +B cubo," for A*+3A2B+3AB2+B32
We copy from Vieta's De emendatione aequationum tradatus secun-
dus (1615) ,3 as printed in 1646, the solution of the cubic x*+W*x = 2Z3 :
"Proponatur A cubus + B piano 3 in A, aequari Z solido 2.
Oportet facere quod propositum est. E quad. +A in E, aequetur B
piano. Vnde B planum ex hujus modi aequationis const itutione, in-
telligitur rectangulum sub duobus lateribus quorum minus est Et
,.,- ,. v . A . ., B planum — E quad. ., . ~
differentia & majore A. igitur — - - ^ -- - - erit A. Quare
Hi
B plano-plano-planum — E quad, in B plano-planum 3+E quad.
quad, in B planum 3 — E cubo-cubo . 5 pi. pi. 3.— B pi. in Eq. 3
- - £ -- 1 — ^—^ - E~ — aequa-
bitur Z solido 2 .
"Et omnibus per E cubum ductis et ex arte concinnatis, E cubi
quad.+Z solido 2 in E cubum, aequabitur B plani-cubo.4
"Quae aequatio est quadrati affirmate affecti, radicem habentis
solidam. Facta itaque reductio est quae imperabatur.
"Confectarium: Itaque si A cubus + B piano 3 in A, aequetur Z
solido 2, & VE plano-plano-plani + Z solido-solido — Z solido,
7^ u T-. B planum — D quad. ., A , ., „
aequetur D cubo. Ii.rgo — £I -- ^ — - - - , sit A de qua quaeritur.
Translation : "Given x*+3B2x = 2Z3. To solve this, let y*+yx = B2.
Since B2 from the constitution of such an equation is understood to be
a rectangle of which the less of the two sides is y, and the difference
B2 — y2
between it and the larger side is x. Therefore ----- ~ — x. Whence
.
y1 y
1 "Cum autem non proponitur utra magnitude sit major vel minor, et tamen
subductio facienda est, nota differentiae est ~ id est, minus incerto: ut propositis
A quadrato et B piano, differentia erit A quadratum :zz B piano, vel B planum
A zn quadrato" (Vieta, Opera mathematica [1646], p. 5).
2 Ibid., p. 17. 3 Ibid., p. 149.
4"£ plani-cubo" should be "B cubo-cubo," and "E cubi quad." should be "E
cubo-cubo."
INDIVIDUAL WRITERS 185
All terms being multiplied by ?/8, and properly ordered, one obtains
7/6+2Z3i/3 = BG. As this equation is quadratic with a positive affected
term, it has also a cube root. Thus the required reduction is effected.
"Conclusion: If therefore x3+3J52a; = 2Z3, and T/B«+Z«-Z* = D*,
B2 — D2.
then — ,. — is x, as required."
The value of x in x?+3B2x = 2Z* is written on page 150 of the
1646 edition thus:
plano-plano-plani+Z solido-solido+Z solido —
* C.VB plano-plano-plani+# solidoTSolido. — Z solido ."
The combining of vinculum and radical sign shown here indicates
the influence of Descartes upon Van Schooten, the editor of Vieta's
collected works. As regards Vieta's own notations, it is evident that
compactness was riot secured by him to the same degree as by earlier
writers. For powers he did not adopt either the Italian symbolism of
Pacioli, Tartaglia, and Cardan or the German symbolism of lludolff
and Stifel. It must be emphasized that the radical sign, as found in
the 1646 edition of his works, is a modification introduced by Van
Schooten. Vieta himself rejected the radical sign and used, instead,
the letter I (latus, "the side of a square") or the word radix. The I
had been introduced by Ramus (§ 322) ; in the Zetetic&rum, etc., of
1593 Vieta wrote 1. 121 for 1/121. In the 1646 edition (p. 400) one
finds V2+V2+l//2+l/2, which is Van Schooten's revision of the
text of Vieta; Vieta's own symbolism for this expression was, in 1593, l
"Radix binomiae 2
(2 (2
+ Radix binomiae \+ radix binomiae \+radice 2 ,"
and in 1595,2
" R. bin. 2+R. bin. 2+R. bin. 2+R. 2. ,"
a notation employed also by his contemporary Adrian Van Roomen.
178. Vieta distinguished between number and magnitude even in
his notation. In numerical equations the unknown number is no longer
represented by a vowel ; the unknown number and its powers are repre-
sented, respectively, by N (numerus), Q (quadratus), C (cubus), and
1 Variorum de rebus mathem. Responsorum liber VIII (Tours, 1593), corollary
to Caput XVIII, p. 12v°. This and the next reference are taken from Tropfke,
op. ciL, Vol. II (1921), p. 152, 153.
2 Ad Problema quod omnibus mathematicis totius orbis conslruendum proposuit
Adrianus Romanus, Francisci Vietae responsum (Paris, 1595), Bl. A IV°.
186 A HISTORY OF MATHEMATICAL NOTATIONS
combinations of them. Coefficients are now written to the left of the
letters to which they belong.
Thus,1 "Si 65C-1QQ, aequetur 1,481,544, fit 1#57," i.e., if
65^-^=1,481,544, then z = 57. Again,2 the "B3 in A quad." occur-
ring in the regular text is displaced in the accompanying example by
"6Q," where £ = 2.
Figure 85 further illustrates the notation, as printed in 1646.
Vieta died in 1603. The De emendations aeqvationvm was first
printed in 1615 under the editorship of Vieta's English friend, Alex-
ander Anderson, who found Vieta's manuscript incomplete and con-
THBORBMA I,
Si A cubtis H- B in A quadr. 3 -* D piano in A , sequetur B cubo z — D
piano in B. A quad. H- BmAi>xquabiturBquad.i — D planer.
Quoniamcnim A quadr. -h B in A i, aecjbatur B quadr. i — D piano. Du&is igicut
omnibus in A. A cubus -t- B in A quad, i, sequabicur B quad, in A z — D piano in A .
Et iifdcm du&is in B. B in A quad, -f B quadr. in A i , gquabuur B cubo i — D piano
in B. lungatur dufta xqualia xqualibus. A cubus •+» B in A quad. 3 H- B quad, in A 2 ,
acquabicur B quad.in A z — D piano in A 4- B cubo i — • D piano in B.
EtdclctaumnqueadfcftioneBquad.in A i, &adacqualitatisordinationcm, tranfla-
raperamithcfin Oplaniin A adfedionc. A cubus -t- B in A quadr. 5 -t- D piano in A,
acquabiturB cuboi — D piano in B. Quodquidcmitafchabct.
FIG. 85. — From Vieta's De emendations aeqvationvm, in Opera mathematica
(1646), p. 154.
taining omissions which had to be supplied to make the tract intelli-
gible. The question arises, Is the notation AT, Q, C due to Vieta or to
Anderson?3 There is no valid evidence against the view that Vieta did
use them. These letters were used before Vieta by Xylander in his
edition of Diophantus (1575) and in Van Schooten's edition4 of the
Ad problema, quod omnibus mathematicis totius orbis construendum
proposuit Adrianus Romanus. It will be noticed that the letter N
stands here for x, while in some other writers it is used in the designa-
tion of absolute number as in Grammateus (1518), who writes our
12x3-24 thus: "12 ter. mi. 24/V." After Vieta N appears as a mark
for absolute number in the Sommaire de Valgebre of Denis Henrion5
1 Vieta, Opera mathematica (1646), p. 223. 2 Op. cit., p. 130.
3 See Enestrom, Bibliotheca mathematica, Vol. XIII (1912-13), p. 166, 167.
4 Vieta, Opera mathematica (1646), p. 306, 307.
6 Denis Henrion, Les qvinze livres des elemens d'Evclide (4th ed.; Paris, 1631),
p. 675-788. First edition, Paris, 1615. (Courtesy of Library of University of
Michigan.)
INDIVIDUAL WRITERS 187
which was inserted in his French edition of Euclid. Henrion did not
adopt Vieta's literal coefficients in equations and further showed his
conservatism in having no sign of equality, in representing the powers
of the unknown by ft, q, c, qq, ft, qc, bfi, qqq, cc, q/3, eft, qqc, etc., and
in using the "scratch method" in division of algebraic polynomials, as
found much earlier in Stifel.1 The one novel feature in Henrion was
his regular use of round parentheses to express aggregation.
ITALIAN: BONAVENTURA CAVALIERI
(1647)
179. Cavalieri's Geometria indivisibilibvs (Bologna, 1635 and
1653) is as rhetorical in its exposition as is the original text of Euclid's
Elements. No use whatever is made of arithmetical or algebraic signs,
not even of + and — , or p and m.
An invasion of German algebraic symbolism into Italy had taken
place in Clavius' Algebra, which was printed at Rome in 1608.
That German and French symbolism had gained ground at the time
of Cavalieri appears from his Exercilationes geometriae sex (1647),
from which Figure 86 is taken. Plus signs of fancy shape appear,
also Vieta's in to indicate "times." The figure shows the expansion
of (a+fr)n for n = 2, 3, 4. Observe that the numerical coefficients are
written after the literal factors to which they belong.
ENGLISH: WILLIAM OUGHTRED
(1631, 1632, 1657)
180. William Oughtred placed unusual emphasis upon the use of
mathematical symbols. His symbol for multiplication, his notation
for proportion, and his sign for difference met with wide adoption in
Continental Europe as well as Great Britain. He used as many as
one hundred and fifty symbols, many of which were, of course, intro-
duced by earlier writers. The most influential of his books was the
Clavis mathematicae, the first edition2 of which appeared in 1631,
later Latin editions of which bear the dates of 1648, 1652, 1667, 1693.
1 M. Stifel, Arithmetica Integra (1544), fol. 239A.
2 The first edition did not contain Clavis mathematicae as the leading words in
the title. The exact title of the 1631 edition was: Arithmeticae in\numeris et sped-\
ebvs institvtio:\Qvae tvm loyislicae, tvm analyli\cae, atqve adeo\(olivs mathematical',
qvasi\davis\esL\ — Ad nobilissimvm spe\ctatissimumque iuvenem I)n. Ovilel\mvm
Howard, Ordinis, qui dici\tur, Balnei Equitem, honoratissimi Dn.\ Thomac,
Comitis Arvndeliae & \ Svrriae, Comitis Mareschal\li Angliae, &c. filium. — \Lon-
dini,\Apud Thomam Harpervm,\ M. DC. xxxi.
188
A HISTORY OF MATHEMATICAL NOTATIONS
A second impression of the 1693 or fifth edition appeared in 1698.
Two English editions of this book came out in 1647 and 1694.
* '''-. ,.a^^'-;""
FIG. 86.— From B. Cavalieri's Exercitationes (1647), p. 268
We shall use the following abbreviations for the designation of
tracts which were added to one or another of the different editions of
the Clavis mathematicae:
Eq. = De Aequationum affectarvm resolvtione in numeris
Eu. = Ekmenti decimi Euclidis declaratio
So. — De Solidis regularibus, tradatus
An. -De Anatotismo, sive usura composita
Fa. = Regula falsae positionis
Ar. = Theorematum in libris Archimedis de sphaera &
cylindro declaratio
Ho. = Horologia scioterica in piano, Geometries delineandi
modus
INDIVIDUAL WRITERS 189
In 1632 there appeared, in London, Oughtred's The Circles of
Proportion, which was brought out again in 1633 with an Addition vnto
the Vse of the Instrument called the Circles of Proportion.1 Another
edition bears the date 1660. In 1657 was published Oughtred's
Trigonometria,2 in Latin, and his Trigonometrie, an English transla-
tion.
We have arranged Oughtred's symbols, as found in his various
works, in tabular form.3 The texts referred to are placed at the head
of the table, the symbols in the column at the extreme left. Each
number in the table indicates the page of the text cited at the head
of the column containing the symbol given on the left. Thus, the nota-
tion : : in geometrical proportion occurs on page 7 of the Clavis of
1648. The page assigned is not always the first on which the symbol
occurs in that volume.
1 In our tables this Addition is referred to as Ad.
2 In our tables Ca. stands for Comones sinuum tangentium, etc., which is the
title for the tables in the Trigonometria.
3 These tables were first published, with notes, in the University of California
Publications in Mathematics, Vol. I, No. 8 (1920), p. 171-80.
190 A HISTORY OF MATHEMATICAL NOTATIONS
181. OUGHTRED'S MATHEMATICAL SYMBOLS
SYMBOLS
MEANINGS
OF
SYMBOLS
Clans mathematica.
ti
£*•<
^§
§
°?
Opu»e. Po*th.r
1677
|
1631
1647
1648
1652
1667
1693
1694
s»S'
S
0[56
0.56
.[58
0,56
OpOOOo
a.6
2.314
2,314
2.314
«.»S
+
P
mo
±
mi
e
1
X
Equal to
Separatrix'
Separatrix
Separatrix3
Separatrix
.00005
Ratio a: 6, or *a— 6
] [Separating5
j | the mantissa
— Characteristic
Arithm. proportion8
0:6, ratio7
Given ratio
Geomet. proportion8
Contin. proportion
Contin. proportion
Geom.9 proportion
( )"
(
( )
( )
( )
( )"
( )"
( )«
Therefore
Addition"
Addition
Addition1'
Subtraction
Plus or minus
Subtraction
Less"
Negative 2
Multiplication15
38
1
34
1
53
1
30
15
1
16
1
73
2
20
3
3
13
235
3
63
29
1
17
5
2
27 '
221
3
7
3
12
3
7
3
7
5
8
25
7
4
3
235
3
Eq.im
&7.187
158
158
22
4n.l62
150
150
21
113
150
21
113
150
21
175
207
32
24
49
11
25
10
19
7
36
3
34
140
87
3
142
42
27
29
114
89
75
53
98
97
116
101
81
5
5
21
5
13
28
8
18
33
7
16
32
7
16
25
7
16
25
7
16
45
40
57
58
115
58
65
107
99
106
58
104
92
104
95
57
52
56
104
95
57
53
92
104
63
63
An. 42
149
119
95
122
95
97
96
35
32
101
101
102
89
151
3
112
2
49
3
3
3
3
57
57
3
3
3
3
4
4
4
4
140
4
4
16
13
99
96
3
2
51
3
57
3
3
106
66
10
56
57
3
53
3
3
17
3
21
96
3
16
4
130
5
97
7
1
10
9
10
1
10
10
5
10
8
37
32
143
INDIVIDUAL WRITERS
182. OUGHTRED'S MATHEMATICAL SYMBOLS— Cont.
191
SYMBOLS
MEANINGS
OP
SYMBOLS
Claris rnathdndticcif
a. co
o0*5
£s
^
0
1
.^
II
£
£
. t--
P
0
1
1631
1647
1648
1652
1667
1693
1694
Hqbq
in
I
a)b(c
flilS
Aq
Ac
Aqq
Aqc
Ace
ABq
[*}.... m
14] .... [10]
a2 . . . . a?
a
Q
(lu
c
Cu
0,0,
oe
D
1,1
L
AL
p
p
R
I*,R
R
0
V
V
v«
VA
X By juxtaposition
Multiplication"
Fraction, division
6-7-a=c
I + J-t
AA
AAA
vl^AA
AAAAA
A/UAAA
ZB*"
4th .... 10th power
4th 10th power
a* . . . . a'
Quaesitum
Square"
Square
Cube
Cube
4th power
5th power
Diameter
Latw, radix19
Angle
Angles
Perimeter
ZA-Aq
Radius
Remainder
Rational
Superficies curva
Root
Square root
Square root
rxi<ufi binomii
7
1
8
10
11
10
12
14
10
10
11
13
11
10
11
14
10
10
11
14
37
10
11
13
13
23
21
96
21
5
16
87
219
9
99
156
104
105
106
17
59
5
50
17
25
41
67
41
7
7
7
7
7
23
11
11
11
11
11
11
37
10
10
10
10
10
11
35
10
10
10
10
10
11
34
35
10
10
10
10
10
11
34
35
10
10
10
10
10
11
14
14
14
55
55
15
53
65
34
52
205
24
38
17
33
16
31
16
57
16
30
16
30
25
47
28
5
100
75
105
53
38
45
33
136
33
33
31
128
61
31
187
30
123
30
30
Eu 21
30
123
30
30
Eu. 21
30
123
30
30
Eu 20
47
175
47
47
28
62
210
210
37
37
121
113
110
110
110
158
139
19
16
192
37
41
120
152
111
134
166
109
126
Eu. 1
AT I
109
128
Eu. 1
Ar 1
109
142
Eu. 1
Ar 1
154
37
32
211
45
35
33
53
49
33
31
48
48
31
30
47
46
30
30
47
46
30
30
47
46
30
47
70
65
47
102
134
96
192 A HISTORY OF MATHEMATICAL NOTATIONS
183. OUGHTRED'S MATHEMATICAL SYMBOLS— Cont.
SYMBOLS
MEANINGS
or
SYMBOLS
Claris mathematicae
£-*
^.^
f
§
t
1
i§
i
&
*" QO
1631
1647
1648
1652
1667
1693
1694
Vr
Vu
VQ</
Vc
V9c
Vcc
V ccc
V cccc
VQU
V[121or1
vQ'j!
r</, re
r, ru
A,E
z
X
z
X
z
9C
c—
--3
tr
_b
C—
r-
rr
Aft
<
>
~cu
"no.
Lotus residui
Sq. rt. of polyno.80
4th root
Cube root
5th root
6th root
9th root
12th root
Square root
12th root
v, f
Square root
Nos., A>E
A+£"
A-E
A*+E>
A>-E>
A3+E>
A*-E»
a-\-e
a—e
a'+fc1
a'-b'
Majus**
Minus
Non majut
Non minus
Minus**
Minus**
Major ratio
Minor ratio
Less than*1
Greater than
Commenaurabilia
Incommensurabilia
35
35
35
37
34
55
52
52
49
52
31
53
47
49
47
49
30
53
46
46
46
48
30
52
46
46
46
48
30
52
48
46
46
48
47
96
69
69
65
69
37
49
37
52
49
48
48
48
69
52
50
49
49
49
69
73
74,96
53
53
53
54
54
94
94
21
21
21
41
41
44
44
33
33
33
33
33
33
33
31
31
31
31
31
31
31
167
30
30
30
30
30
30
30
Eu. 1
Eu. 1
30
30
30
30
30
30
30
EuA
EuA
30
30
30
30
30
30
30
EuA
EuA
47
47
47
47
47
47
47
87
87
87
98
99
19
16
16
167
167
166
166
166
166
Eu.2
Eu.2
145
EuA
Eu.l
Eu.l
Eu.2
Eu.2
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
Ho, 17
Ho. 17
I
Ho. 30
166
166
Ho. 31
Eu. 1
Eu. 1
Ho. 29
EuA
EuA
EuA
EuA
11
6
4
4
166
166
Eu.l
Eu.l
EuA
EuA
Eu.l
EuA
1
INDIVIDUAL WRITERS
184. OUGHTRED'S MATHEMATICAL SYMBOLS— Cont.
193
SYMBOLS
MEANINGS
or
SYMBOLS
Claris mathematicae
§"§?
£-«
^s"
ai
i
fri*"
i
il
•S
1
i?""1
O
1631
1647
1648
1652
1667
1693
1694
*
V
V
nT
r
<r
T
eim
cr
_o
£=
SSJ
0
o
A
4
v^*\
11
log
log:^:
S
t
se
8V
t ver
sin : com
SCO
too
86 CO
sin
tan
sec
sec:parall
Comment, potentia
Incommens. potentia
Rationale
Irrationale
Medium
Line, cut extr. and mean
ratio
Major ejus portio
Minor ejus portio
Simile
Proxime majus
Proxime minus
Aequale vel minut
Aeqitale vel ma jus
Rectangulum
Quadratum
Trianguhan
Latus, radix
Media proportion
Differentia^*
Parallel
Logarithm
Log. of square
Sine"
Tangent
Secant
•Sinus versut
Sinus versus™
Sine complement
Cosine
Cotangent
Cosecant
Sine
Tangent
Secant
Sum of secants
166
166
166
166
166
166
166
166
166
166
166
166
166
167
167
167
167
167
Eu. 1
Eu. 1
Eu. 1
Eu. 1
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
Eu.2
Eu.2
Eu.2
Eu.2
Eu.2
Eu.2
Eu. 1
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
Eu.2
Eu.2
Eu.2
Eu.2
Eu.2
Eu.2
Eu. I
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
EuA
Eu.l
EuA
EuA
EuA
EuA
EuA
EuA
EuA
Eu I
33
51
17
149
147
197
172
135
Ho. 29
158
127
150
122
150
122
122
122
207
174
17
96
96
5
3
14
172
174
Ho. 29
76
107
99
98
98
98
140
5
Ad. 69
96
96
3
3
4
35
Ca. 3
174
ffo. 41
ffo. 41
Ho. 42
Ad. 69
Ad. 69
37
Ad. 41
Ad. 41
194 A HISTORY OF MATHEMATICAL NOTATIONS
185. OUGHTRED'S MATHEMATICAL SYMBOLS— Cont.
SYMBOLS
MEANINGS
or
SYMBOLS
Clam Mathematical
g-w
£2
^8
w'S2
o
1
&-.
£
j|co
.4
1
1631
1647
1648
1652
1667
1693
1694
tang
C
Cent
Ho. ' "
9
V
T
7
M
m
m
Gr.
rain.
JNI
!— 1
Lo
1
D
Tri, tri
M
X
Z cru
Zcrur
Xcrw
Xcrur
A
L
'1
i
CO
T
X
Z
Tangent
.01 of a degree
.01 of a degree
Degr., min., sec.
Hours, min., sec.
180 - angle
Equal in no. of degr.
ir=3.1418
Canceled*
Mean proportion
Minus
34 3 4 9*8
-X-=2, --*--=-
23 238
Degree
Minute
Differentia
Aequalia lempore
Logarithm
Separatrix
Differentia
Triangle
Cent, minute of arc
Multiplication"
(Z sum, X diff .
of sides of
rectangle"
or triangle
Unknown
Altit. frust. of pyramid
or cone
Altit. of part cut off
First term ^
Last term .2
No. of terms 1 &
Common differ. I g.
Sum of all termsj °
Ho. 29
Ho. 41
Ho. 41
Ho. 42
12
235
236
235
21
20
21
20
21
32
66
07
36
2
6
68
72
100
69
94
66
90
Ar. 1
66
90
Ar. 1
66
90
Ar. 1
99
131
20
32
20
30
19
29
Ho. 23
29
Ho. 23
29
19
45
29
235
Ad. 19
134
68
Ca.2
244
19
237
24
Ca.2
76
191
Eu. 26
70
69
5
17
101
16
16
17
16
38
77
77
13
53
109
109
85,18
85,18
85
85
85,18
51
101
101
80,17
80,17
80
80
80,17
50
99
99
78,16
78,16
78
78
78,16
50
99
99
78,16
78,16
78
78
78,16
50
99
99
78,16
78,16
78
78
78,16
72
141
142
116,26
116,26
116
116
116,26
113
84
19
30,116
30, 116
11
no
30,116
19
INDIVIDUAL WRITERS 195
186. Historical notes1 to the tables in §§ 181-85:
1. All the symbols, except "Log," which we saw in the 1660 edition of the
Circles of Proportion, are given in the editions of 1632 and 1633.
2. In the first half of the seventeenth century the notation for decimal frac-
tions engaged the attention of mathematicians in England as it did elsewhere
(see §§ 276-89). In 1608 an English translation of Stevin's well-known tract was
brought out, with some additions, in London by Robert Norton, under the title,
Disme: The Art of Tenths, or, Decimall Arilhmetike (§ 276). Steviri's notation is
followed also by Henry Lyte in his Art of Tens or Decimall Arith?nelique (London,
1619), and in Johnsons Arithmetick (2d ed.; London, 1633), where 3576.725 is
123
written 3576|725. William Purser in his Compound Interest and Annuities (London,
1634), p. 8, uses the colon (:) as the separator, as did Adrian us Metius in his
Geometnae practicae pars I et II (Lvgd., 1625), p. 149, and Rich. Balam in his
Algebra (London, 1653), p. 4. The decimal point or comma appears in John
Napier's Rabdologia (Edinburgh, 1617). Oughtred's notation for decimals must
have delayed the general adoption of the decimal point or comma.
3. This mixture of the old and the new decimal notation occurs in the Key of
1694 (Notes) and in Gilbert Clark's Oughtredus explicatus2 only once; no reference
is made to it in the table of errata of either book. On Oughtred's Opuscula mathc-
matica hactenus inedita, the mixed notation 128,57 occurs on p. 193 fourteen times.
Oughtred's regular notation 128 [57 hardly ever occurs in this book. We have seen
similar mixed notations in the Miscellanies: or Mathematical Lucubrations, of Mr.
Samuel Foster, Sometime publike Professor of Astronomic in Gresham Coltedgc,
in London, by John Twysden (London, 1659), p. 13 of the "Observations eclipsi-
um"; we find there 32.466, 31.008.
4. The dot (.), used to indicate ratio, is not, as claimed by some writers, used
by Oughtred for division. Oughtred does not state in his book that the dot (.)
signifies division. We quote from an early arid a late edition of the Clavis. He
says in the Clavis of 1694, p. 45, and in the one of 1648, p. 30, "to continue ratios
is to multiply them as if they were fractions." Fractions, as well as divisions, are
indicated by a horizontal line. Nor does the statement from the Clavis of 1694,
p. 20, and the edition of 1648, p. 12, "In Division, as the Divisor is to Unity, so is
the Dividend to the Quotient," prove that he looked upon ratio as an indicated
division. It does not do so any more than the sentence from the Clavis of 1694,
and the one of 1648, p. 7, "In Multiplication, as 1 is to either of the factors, so is
the other to the Product," proves that ho considered ratio an indicated multiplica-
tion. Oughtred says (Clavis of 1694, p. 19, and the one of 1631, p. 8): "If Two
Numbers stand one above another with a Line drawn between them, 'tis as much
12 5
as to say, that the upper is to be divided by the under; as -j- and -^ •"
1 N. 1 refers to the Circles of Proportion. The other notes apply to the super-
scripts found in the column, "Meanings of Symbols."
2 This is not a book written by Oughtred, but merely a commentary on the
Clavis. Nevertheless, it seemed desirable to refer to its notation, which helps to
show the changes then in progress.
196 A HISTORY OF MATHEMATICAL NOTATIONS
In further confirmation of our view we quote from Oughtred's letter to W.
Robinson: ''Division is wrought by setting the divisor under the dividend with a
line between them."1
5. In Gilbert Clark's Oughtredus explicatus there is no mark whatever to sepa-
rate the characteristic and mantissa. This is a step backward.
6. Oughtred's language (Clavisoi 1652, p. 21) is: "Ut 7.4: 12.9 vel 7.7-3: 12.12
—3. Arithmetic^ proportionates sunk" As later in his work he does not use arith-
metical proportion in symbolic analysis, it is not easy to decide whether the sym-
bols just quoted were intended by Oughtred as part of his algebraic symbolism or
merely as punctuation marks in ordinary writing. Oughtred's notation is adopted
in the article "Caractere" of the Encyclopedic methodique (mathematiques) , Paris:
Liege, 1784 (see § 249).
7. In the publications referred to in the table, of the years 1648 and 1694, the
use of : to signify ratio has been found to occur only once in each copy; hence we
are inclined to look upon this notation in these copies as printer's errors. We are
able to show that the colon (:) was used to designate geometric ratio some years
before 1657, by at least two authors, Vincent Wing the astronomer, and a school-
master who hides himself behind the initials "R.B." Wing wrote several works.
8. Oughtred's notation A.B::C.D, is the earliest serviceable symbolism for
proportion. Before that proportions were either stated in words as was customary
in rhetorical modes of exposition, or else was expressed by writing the terms of the
proportion in a line with dashes or dots to separate them. This practice was in-
adequate for the needs of the new symbolic algebra. Hence Oughtred's notation
met with ready acceptance (see §§ 248-59).
9. We have seen this notation only once in this book, namely, in the expres-
sion R.S. =3.2.
10. Oughtred says (Clavis of 1694, p. 47), in connection with the radical sign,
"If the Power be included between two Points at both ends, it signifies the uni-
versal Root of all that Quantity so included; which is sometimes also signified by
b and r, as the i/b is the Binomial Root, the -\/r the Residual Root." This notation
is in no edition strictly adhered to; the second : is often omitted when all the terms
to the end of the polynomial are affected by the radical sign or by the sign for a
power. In later editions still greater tendency to a departure from the original
notation is evident. Sometimes one dot takes the place of the two dots at the end;
sometimes the two end dots are given, but the first two are omitted; in a few
instances one dot at both ends is used, or one dot at the beginning and no symbol
at the end; however, these cases are very rare and are perhaps only printer's errors
We copy the following illustrations:
Q : A -E: est Aq-2AE+Eq, for (A -E? = A2-2A#-f-#2 (from Chans of 1631, p.
45)
, for
(from Clavis of 1648, p. 106)
: BA+ CA =BC+ Z), for ^(BA+CA) = BC+D (from Clavis of 1631, p. 40)
AB . , ABq CXS . ( AE . I /ZE2 CXS\ A ,. ni . f
£i -- R~ : =4., for -13-+ <v/ (-4 --- --) "A. (from Clavis of 1652,
p. 95)
1 Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. I
(1841), Letter VI, p. 8.
INDIVIDUAL WRITERS 197
Q.Hc+Ch : for (Hc+Ch)* (from Claris of 1652, p. 57)
Q.A-X=, for (A-X)*= (from Clavis of 1694, p. 97)
--fr.tt. ~!-CD.=A, for|+J(^-CzA =A (from Oughtredus explicates
[1682], p. 101)
11. These notations to signify aggregation occur very seldom in the texts re-
ferred to and may be simply printer's errors.
12. Mathematical parentheses occur also on p. 75, 80, and 117 of G. Clark's
Oughtredus explicates.
13. In the Clavis of 1631, p. 2, it says, "Signum additionis siue affirmationis,
est+plus" and "Signum subductionis, siue negationis est— minus." In the edition
of 1694 it says simply, "The Sign of Addition is + more" and "The Sign of Sub-
traction is — less," thereby ignoring, in the definition, the double function played
by these symbols.
14. In the errata following the Preface of the 1694 edition it says, for "more
or mo. r. [ead] plus or pi."', for less or le. r.[ead] minus or mi."
15. Oughtred's Clavis mathematicae of 1631 is not the first appearance of X
as a symbol for multiplication. In Edward Wright's translation of John Napier's
Descriptio, entitled A Description of the Admirable Table of Logarithms (London,
1618), the letter "X" is given as the sign of multiplication in the part of the book
called "An Appendix to the Logarithms, shewing the practise of the calculation of
Triangles, etc."
The use of the letters x and X for multiplication is not uncommon during the
seventeenth and beginning of the eighteenth centuries. We note the following
instances: Vincent Wing, Doctrina theorica (London, 1656), p. 63; John Wallis,
Arithmetica infinitorum (Oxford, 1655), p. 115, 172; Moore's Arithmelick in two
Books, by Jonas Moore (London, 1660), p. 108; Antoine Arnauld, Novveavx elemens
de geometrie (Paris, 1667), p. 6; Lord Brounker, Philosophical Transactions, Vol.
II (London, 1668), p. 466; Exercitatio geometrica, auctore Laurenlio Lorenzinio,
Vincentii Viviani discipulo (Florence, 1721). John Wallis used the X in his
Elenchus geometriae Hobbianae (Oxoniae, 1655), p. 23.
16. in as a symbol of multiplication carries with it also a collective meaning;
for example, the Clavis of 1652 has on p. 77, "Erit \Z + \B in \Z - \B = \Zq - $Bq."
17. That is, the line AB squared.
18. These capital lettprs precede the expression to be raised to a power. Sel-
dom are they used to indicate powers of monomials. From the Clavis of 1652, p. 65,
we quote:
"C : A +E : +Eq=2Q : \A +E : +2Q4A ,"
i.e.,
19. L and I stand for the same thing, "side" or "root," I being used generally
when the coefficients of the unknown quantity are given in Hindu-Arabic numerals,
so that all the letters in the equation, viz., I, q, c, qq, qc, etc., are small letters. The
Clavis of 1694, p. 158, uses L in a place where the Latin editions use I.
20. The symbol i/u does not occur in the Clavis of 1631 and is not defined in
the later editions. The following throws light upon its significance. In the 1631
edition, chap, xvi, sec. 8, p. 40, the author takes i/qBA+ B = CA, gets from it
y qBA =CA-B, then squares both sides and solves for the unknown A. He passes
198 A HISTORY OF MATHEMATICAL NOTATIONS
next to a radical involving two terms, and says: "Item \/q vniuers : BA+ CA : —
D — BC : vel per transpositionem 1/9 : BA-+-CA =BC+D"'r he squares both sides
and solves for A. In the later editions he writes "|/V in place of "i/g
vniuers : "
21. The sum Z = A+E and the difference X=A— E are used later in imita-
tion of Oughtred by Samuel Foster in his Miscellanies (London, 1659), "Of
Projection/' p. 8, and by Sir Jonas Moore in his Arithmelick (3d ed.; London, 1688),
p. 404; John Wallis in his Operum mathematicorum pars prima (Oxford, 1657),
p. 169, and other parts of his mathematical writings.
22. Harriot's symbols > for "greater" and < for "less" were far superior to
the corresponding symbols used by Oughtred.
23. This notation for "less than" in the Ho. occurs only in the explanation of
"Fig. Efi." In the text (chap, ix) the regular notation explained in En. is used.
24. The symbol GO so closely resembles the symbol <v> which was used by
John Wallis in his Operum mathematicorum pars prima (Oxford, 1657), p. 208,
247, 334, 335, that the two symbols were probably intended to be one and the
same. It is difficult to assign a good reason why Wallis, who greatly admired
Oughtred and was editor of the later Latin editions of his Clavis mathematicae,
should purposely reject Oughtred's GO and intentionally introduce ~ as a substi-
tute symbol.
25. Von Braunmiihl, in his Geschichte dcr Trigonometric (2. Teil; Leipzig,
1903), p. 42, 91, refers to Oughtred's Trigonomelria of 1657 as containing the
earliest use of abbreviations of trigonometric functions and points out that a half-
century later the army of writers on trigonometry had hardly yet reached the
standard set by Oughtred. This statement must be modified in several respects
(see §§ 500-526).
26. This reference is to the English edition, the Trigonometric of 1657. In the
Latin edition there is printed on p. 5, by mistake, 8 instead of s versus. The table of
errata makes reference to this misprint.
27. The horizontal line was printed beneath the expression that was being
crossed out. Thus, on p. 68 of the Clavis of 1631 there is:
BGqq-BGqX2BK XBD+BKq XBDg
= BGqXBDq+BGq X BKg -BGqX2BKXBD + BGq X 4CAq.
28. This notation, says Oughtred, was used by ancient writers on music, who
"are wont to connect the terms of ratios, either to be continued" as in |X| = 2,
"or diminished" as in \ -s- j = f .
29. See n. 15.
30. Cru and crur are abbreviations for crurum, side of a rectangle or right tri-
angle. Hence Z cru means the sum of the sides, X crut the difference of the sides.
187. Oughtred's recognition of the importance of notation is
voiced in the following passage:
". . . . Which Treatise being not written in the usuall synthetical
manner, nor with verbous expressions, but in the inventive way of
Analitice, and with symboles or notes of things instead of words,
seemed unto many very hard; though indeed it was but their owne
diffidence, being scared by the newness of the delivery; and not any
INDIVIDUAL WRITERS 199
difficulty in the thing it selfe. For this specious and symbolicall man-
ner, neither racketh the memory with multiplicity of words, nor
chargeth the phantasie with comparing and laying things together;
but plainly presenteth to the eye the whole course and processe of
every operation and argumentation/'1
Again in his Circles of Proportion (1632) , p. 20 :
"This manner of setting downe theoremes, whether they be Pro-
portions, or Equations, by Symboles or notes of words, is most excel-
lent, artificiall, and doctrinall. Wherefore I earnestly exhort every
one, that desireth though but to looke into these noble Sciences
Mathematicall, to accustome themselves unto it: and indeede it is
easie, being most agreeable to reason, yea even to sence. And out of
this working may many singular consectaries be drawne: which
without this would, it may be, for ever lye hid."
ENGLISH: THOMAS HARRIOT
(1631) '
188. Thomas Harriot's Artis analyticae praxis (London, 1631)
appeared as a posthumous publication. He used small letters in place
of Vieta's capitals, indicated powers by the repetition of factors, and
invented > and < for "greater" and "less."
Harriot used a very long sign of equality =. The following quo-
tation shows his introduction of the now customary signs for "greater"
and "smaller" (p. 10):
"Comparationis signa in sequentibus vsurpanda.
Aequalitatis • ut a = b. significet a acqualem ipi b.
Maioritatis :r> ut a ;>- 6. significet a maiorem quam b.
Minoritatis <d.ut a -<: b significet a minorern quam ft."
Noteworthy is the notation for multiplication, consisting of a
vertical line on the right of two expressions to be multiplied together
of which one is written below the other; also the notation for complex
fractions in which the principal fractional line is drawn double. Thus
(p. 10):
ac
ado
b
b b
aaa
b aaa
••ac ,
d bd •*
1 William Oughtred, The Key of the Mathematics (London, 1647), Preface.
200 A HISTORY OF MATHEMATICAL NOTATIONS
Harriot places a dot between the numerical coefficient and the
other factors of a term. Excepting only a very few cases which seem
to be printer's errors, this notation is employed throughout. Thus
(p. 60):
"Aequationis aaa— 3.baa+3.bba=+2.bbb est 2.6, radix
radici quaesititiae a. aequalis ."
Probably this dot was not intended as a sign of multiplication, but
simply a means of separating a numeral from what follows, according
to a custom of long standing in manuscripts and early printed books.
On the first twenty-six pages of his book, Harriot frequently
writes all terms on one side of an equation. Thus (p. 26) :
"Posito igitur cdf=aaa. est aaa— cdf\ = 0
a+b I
Est autem ex genesi aaa—cdf\ = aaaa-t-baaa—cdfa—bcdf.
a+b \
quae est aequatio originalis hie designata.
Ergo .... aaaa+baaa—cdfa—bcdf. = 0 ."
Sometimes Harriot writes underneath a given expression the result
of carrying out the indicated operations, using a brace, but without
using the regular sign of equality. This is seen in Figure 87. The
first equation is 52=— 3a+aaa, where the vowel a represents the
unknown. Then the value of a is given by Tartaglia's formula, as
^26+ V 675+ ^26 -1/675 = 4. Notice thal^VS.)" indicates that
the cube root is taken of the binomial 26+1/675.
In Figure 88 is exhibited Harriot's use of signs of equality placed
vertically and expressing the equality of a polynomial printed above a
horizontal line with a polynomial printed below another horizontal
line. This exhibition of the various algebraic steps is clever.
FRENCH: PIERRE HERIGONE
(1634, 1644)
189. A full recognition of the importance of notation and an
almost reckless eagerness to introduce an exhaustive set of symbols
is exhibited in the Cursus mathematicus of Pierre H£rigone, in six
volumes, in Latin and French, published at Paris in 1634 and, in a
second edition, in 1644. At the beginning of the first volume is given
INDIVIDUAL WRITERS
201
40:
^^*£!!3*£!^£j£
FIG. 87. — From Thomas Harriot's Ar/is analyticae praxis (1631), p. 101
Si daripoffit radix alinua arquatioms radio 4* xquaHs,<iux radicibu$ ^ e. d. ina:-
qualis fit , cfto ilia / fiue alia quxcunquc*
Pofitoigiwr/=^ mtffff—tfff+tttf
+ffff—
Hoceft
+ffff—'ff/+«*ff—*fff
—t, dff
Ergo
QiJod eft contra Lemmatishypothcfui.
Non eft igitur /=='«. vtcratpofltum. Quod de alia quacnncjue ex fimilidc-
duftione demonftrandom eft.
Fia. 88. — From Thomas Harriot's Artis analytical praxis (1631), p. 65
202 A HISTORY OF MATHEMATICAL NOTATIONS
an explanation of the symbols. As found in the 1644 edition, the list
is as follows:
+ plus is, signifie le plurier
~ minus 2|2 aequalis
•~: differentia 3|2 maior
< inter se, entrflks 2|3 minor
4 n in, en i tertia pars
4 ntr. inter, entre I quarta pars
.11 vel, ou I duae tertiae
TT, ad, a a,b, 11 ab rectangulum quod sit
5< pentagonum, penta- ductu A in B
gone • est punctum
6< hcxagonum — est recta linea
l/»4< latus quadrati <, Z est angulus
l/-5< latus pentagon! _J est angulus rectus
a2 A quadratum O est circulus
a3 A cubus *3> & est pars circumfer-
a4 A quadrato-quadratu. entiae circuli
et sic infinitum. Q, o est segmentu circuli
= parallela A est triangulum
JL perpendicularis D est quadratum
• • est nota genitini, sig- a est rectangulum
nifie (de) <3> est parallelogrammum
; est nota numeri plural- <0> piped, est parallelepipedum
In this list the symbols that are strikingly new are those for equality
and inequality, the ^ as a minus sign, the — being made to represent
a straight line. Novel, also, is the expression of exponents in Hindu-
Arabic numerals and the placing of them to the right of the base, but
not in an elevated position. At the beginning of Volume VI is given a
notation for the aggregation of terms, in which the comma plays a
leading role:
"O a2~5a+6, a~4: virgula, la virgule, dis-
tinguit multiplicatorem a~4 d multiplicado
Ergo 0 a 5+4+3, 7~3:~10, est 38."
Modern: The rectangle (a2 — 5a+6) (a— 4) ,
Rectangle (5+4+3) (7-3) -10 = 38 .
"hg TT ga 2|2 hb <* bd, signifi. HO est ad GA, vt
HB ad BD ."
INDIVIDUAL WRITERS
203
''
'-'-* J~ *4i u^~i' J-
foufticnt l'
,
FIG. 89. — From P. Herigone, Cursuv mathematicus, Vol. VI (1641) ; proof oi the
Pythagorean theorem.
204 A HISTORY OF MATHEMATICAL NOTATIONS
Modern: hg : ga = hb : bd .
"l/«16+9 est 5, se pormoit de serirc plus dis-
tinctement ainsi ,
l/-(16+9) 111AT6+9, est 5:i/-9, +4, sont
7: j/-9, +|/-4 sont 5: "
Modern: y • 16+9 is 5, can be written more clearly thus,
V7- (16+9) or v7- 16+9, is 5; ]/-9, +4, are 7;
1/-9, + i/-4 are 5 .
FRENCH: JAMES HUME
(1635, 1636)
190. The final development of the modern notation for positive
integral exponents took place in mathematical works written in
French. Hume was British by birth. His Le traite d'algebre (Paris,
1635) contains exponents and radical indexes expressed in Roman
numerals. In Figure 90 we see that in 1635 the plus (+) and minus
( — ) signs were firmly established in France. The idea of writing
exponents without the bases, which had been long prevalent in the
writings of Chuquet, Bombelli, Stevin, and others, still prevails in the
1635 publication of Hume. Expressing exponents in Roman symbols
made it possible to write the exponent on the same line with the coeffi-
cient without confusion of one with the other. The third of the ex-
amples in Figure 90 exhibits the multiplication of 8x2+3x by IQx,
yielding the product 80x3+30x2.
The translation of part of Figure 91 is as follows: "Example: Let
there be two numbers 1/9 and 1/8, to reduce them it will be necessary
to take the square of 1/8, because of the II which is with 9, and the
square of the square of 1/9 and you obtain 1/6561 and 1/64
f8 to 1/64
1/9 to 1/729
1/3 to 1/8 [should be 1/9]
1/2 to 1/9 [should be 1/8]
1/3 to 1/9
1/2 to V32 ."
The following year, Hume took an important step in his edition of
Ualgebre de Viete (Paris, 1636), in which he wrote Aiu for A3. Except
for the use of the Roman numerals one has here the notation used by
Descartes in 1637 in his La geometric (see § 191).
INDIVIDUAL WRITERS
205
FRENCH : RENE DESCARTES
(1637)
191. Figure 92 shows a page from the first edition of Descartes'
La gtomttrie. Among the symbolic features of this book are: (1) the
use of small letters, as had been emphazised by Thomas Harriot;
I ^ « ' Jrr ^ f* ' * ' *l 1* f>^S-J
' '
FIG. 90. — Roman numerals for unknown numbers in James Huine, Algbbre
(Paris, 1635).
(2) the writing of the positive integral exponents in Hindu-Arabic
numerals and in the position relative to the base as is practiced today,
206
A HISTORY OF MATHEMATICAL NOTATIONS
except that aa is sometimes written for a2; (3) the use of a new sign of
equality, probably intended to represent the first two letters in the
word aequalis, but apparently was the astronomical sign, & taurus,
tf^V f" V «, >**. Eir
Lv?,,'1 lt»rt$efoh
1'jG. 91.— -Radicals in Junics Hume, Algvbrc (1635)
INDIVIDUAL WRITERS
207
.
jtfteFW^la^
T'W ^-J ^s%'^^>l& $y^ffa#WS'<;\'&fck WV" ">' '^-- '•;'•
"MV& ^;V^;'ji4^;?^iHe;|t;»^;Wtt^^ ri.-ft -i'W^
FIG. 92. — A page from Ren6 Descartes, Lo gtamttrie (1637)
208 A HISTORY OF MATHEMATICAL NOTATIONS
placed horizontally, with the opening facing to the left; (4) the uniting
of the vinculum with the German radical sign j/, so as to give i/*"~~,
an adjustment generally used today.
The following is a quotation from Descartes' text (ed., Paris,
1886, p. 2): "Mais souvent on n'a pas besoin de tracer ainsi ces lignes
sur le papier, et il suffit de les designer par quelques lettres, chacune
par une seule. Comme pour ajouter le ligne BD a GH, je nomme
Pune a et Pautre 6, et £cris a+b; et a— b pour soustraire b de a; et ab
pour les multiplier Pune par Pautre; et T pour diviser a par b; et aa ou
a2 pour multiplier a par soi-meme; et a3 pour le multiplier encore une
fois par a, et ainsi a Pinfini. "
The translation is as follows: "But often there is no need thus to
trace the lines on paper, and it suffices to designate them by certain
letters, each by a single one. Thus, in adding the line BD to GH, I
designate one a and the other 6, and write a+b; and a— 6 in sub-
tracting b from a; and ab in multiplying the one by the other; and j- in
dividing a by 6; and aa or a2 in multiplying a by itself; and a3 in
multiplying it once more again by a, and thus to infinity/'
ENGLISH: ISAAC BARROW
(1655, 1660)
192. An enthusiastic admirer of Oughtred's symbolic methods
was Isaac Barrow,1 who adopted Oughtred's symbols, with hardly
any changes, in his Latin (1655) and his English (1660) editions of
Euclid, Figures 93 and 94 show pages of Barrow's Euclid.
ENGLISH: RICHARD RAWLINSON
(1655-68)
193. Sometime in the interval 1655-68 Richard Rawlinson, of
Oxford, prepared a pamphlet which contains a collection of litho-
graphed symbols that are shown in Figure 95, prepared from a crude
freehand reproduction of the original symbols. The chief interest lies
in the designation of an angle of a triangle and its opposite side by the
same letter — one a capital letter, the other letter small. This simple
device was introduced by L. Euler, but was suggested many years
earlier by Rawlinson, as here shown. Rawlinson designated spherical
1 For additional information on his symbols, see §§ 456, 528.
INDIVIDUAL WRITERS
209
triangles by conspicuously rounded letters and plane triangles by
letters straight in part.
i' ' , - -<i - ' ~
&raW«»:'fit tiM1' '?^
p^p „ r^.w j _T _ *f"(L T W » r^
ib'ii.tiffOTMafflfK
ijj^ik^ ^ i1 j - _ .,.,.-•-«--"•-- I- —
*y8p ' At>l4<i)w4*«w ABo^>fif«ifl'» rfe/wwr /g»r4
^^.fe^W11*/^? * w^B-AD, DB/»^if«rf/*f««-
**r"^^*uJ^"(3'fl^w ^^M> <S*^ » SM/«!' AB taxitftHt
'?*~H~~vtW$to ?**«*«>*»• <5K ,' ««K«r4(W fj? qtoetMuto
'&'-. * ?JBgtM^fe f D' JM tevtu&ae tammta^«m:
5*rsT#.ji !*w-l^fcfe .^/ff* : e^.it?* gj;
l**fiW:^^^^V^(i^«r4l^^' fW««' <w<«»» /««*
• 0 »*& X,;^' W«>" fWtlf i/^f.^ ,*3^ «*^f
• - .. i • . ; M , ,,j!^liu.^- A- *«i, ^ llWi)flr<a( AB q$Mt*tt
^'••k$$y*,^&: Atypfrim*
^14'A^l^1,
1^€^I^'
% ' ' r i i K*i t*i k, I IT- * *_.,tr
FIG. 93. — Latin edition (1655) of Barrow's Euclid. Notes by Isaac Newton.
(Taken from Isaac Newton: A Memorial Volume [ed. W. J. Greenstreet; London,
1927], p. 168.)
210
A HISTORY OF MATHEMATICAL NOTATIONS
DF :: CB FH,vth3t »,
••
,:J¥*:^':'; ;u ,f\ '| y1' ; /y^y/v^L; ^y^^V^'^r' , V
J AS^^AIJi^ iJ^Jl^'"^^^ ^ |f[ ! -' ' '' ^
^W^CBLF^fU^f^'^^H:;,
•-- ---tin ^;p«tt^3fc*|;%' ^:; 4
FIG. 94. — English edition of Isaac Barrow's Euclid
INDIVIDUAL WRITERS
211
SWISS! JOHANN HEINRICH RAHN
(1659)
194. Rahn published in 1659 at Zurich his Teutsche Algebra,
which was translated by Thomas Brancker and published in 1668 at
London, with additions by John Pell. There were some changes in
the symbols as indicated in the following comparison :
Meaning
German Edition, 1659
English
Edition, 1668
1 Multiplication
* (p 7)
Same
(r> ft)
2 d~\-b times a — b
2 + hl (P H)
Same
(p 12)
3. Division
4. Cross-multiplication
5 Involution
-J- (P. 8)
*X (p. 25)
Archimedean spi-
Same
*X
Ligature of
(p. 7)
(p. 23)
omicron and
6 Evolution
ral (Fig. 96) (p. 10)
Ligature of two
sigraa (Fig.
Same
97) (p. 9)
fn 0^
7. Erf Ull ein quadrat \
Compleat the square /
epsilons(Fig.96)(p. 11)
#D (P 16)
CD
(P- 14)
8 Sixth root
/ /aaa — V«
cubo -cubic k V
o/aaa» V« (p. 32)
9 Therefore
y/' \ aa = Vc.a (p. 34)
.'. (usually) (p 53)
cubo-cubick V
'.' (usually)
of aa«=» jc.a
(n 37^
10. Impossible (absurd)
11. Equation expressed in an-
other way
2 (P- 01)
(p. 67)
01
Same
(P. 48)
(p 64)
12. Indeterminate, "liberty of as-
suming an equation"
13. Nos. in outer column refer-
ring to steps numbered in
middle column
14. Nos in outer column not re-
ferring to numbers in middle
column
(*) (p. 89^
1,' 2,' 3*. etc. (p. 3)
1 2, 3, etc (p 3)
Same
1, 2, 3, etc.
1, 2, 3, etc
(P. 77)
(p. 3)
(p 3)
REMARKS ON THESE SYMBOLS
No. 1. — Rahri's sign * for multiplication was used the same year as Brancker's translation, by
N. Mercator, in his Louarithrnotechnia (London, 1668), p. 28.
No. 4. — If the lowest common multiple of abc and ad is required, Rahn writes —-T •»—.; then
— -,'-*X-7 yields abed in each of the two cross-multiplications.
ad d
No. 8. — Hahn's and Brancker's modes of indicating the higher powers and roots differ in
principle and represent two different procedures which had been competing for supremacy for several
centuries. Rahn's V^- means the sixth root, 2X3 = 6, and represents the Hindu idea. Brancker's
cubo-cubick root means the "sixth root," 3+3 = 6, and represents the Diophantine idea.
No. 9. — In both editions occur both /. and v, but /. prevails in the earlier edition; v prevails in
the later.
No. 10. — The symbols indicate that the operation is impossible or, in case of a root, that it is
imaginary.
Wo. 11. — The use of the comma is illustrated thus: The marginal column (1668, p. 54) gives
"6, 1," which means that the sixth equation "Z = A" and the first equation "A=«6" yield Z=»6.
No. 12. — For example, if in a right triangle h, b, c, we know only b— c, then one of the three
sides, say c, is indeterminate.
Page 73 of Rahn's Teutsche Algebra (shown in Fig. 96) shows:
(1) the first use of -5- in print, as a sign of division; (2) the Archimede-
an spiral for involution; (3) the double epsilon for evolution; (4) the
212
A HISTORY OF MATHEMATICAL NOTATIONS
use of capital letters J5, D, E, for given numbers, and small letters
a, 6, for unknown numbers; (5) the ^ for multiplication; (6) the first
use of .'. for "therefore"; (7) the three-column arrangement of which
the left column contains the directions, the middle the numbers of
^ *y b , r
i» ^ / " /
1'iG. 95.— Freehand reproduction of Richard Rawlinson's symbols
the lines, the right the results of the operations. Thus, in line 3,
we have "line 1, raised to the second power, gives aa+2ab-\-bb=DD."
ENGLISH: JOHN WALLIS
(1655, 1657, 1685)
195. Wallis used extensively symbols of Oughtred and Harriot,
but of course he adopted the exponential notation of Descartes (1637).
Wallis was a close student of the history of algebra, as is illustrated
INDIVIDUAL WRITERS
213
by the exhibition of various notations of powers which Wallis gave in
1657. In Figure 98, on the left, are the names of powers. In the first
column of symbols Wallis gives the German symbols as found in
Stifel, which Wallis says sprang from the letters r, z, c, J, the first
b' t*--1
iflY'^V^'' *r'fT^fl' u?'^; ^fy^W^'''1^ $X'''$**'1'-.i ?* '^'"^ ^ r'l; '^
- "'p^-'r 'LjX'V^ ^ ;'j'^; /^v'"^, 'i n 4'1 ;•'- ;!;.'--' ^ ^ --<l :, ^
'•7^^^''':''Mi^_±i^^:l^.:
FIG. 90.— From Rahn, Teutsche Algebra (1659)
letters of the words res, zensus, cubus, sursolidus. In the second column
are the letters R, Q, C, S and their combinations, Wallis remarking
that for R some write N; these were used by Vieta in numerical equa-
tions. In the third column are Vieta's symbols in literal algebra, as
abbreviated by Oughtred; in the fourth column Harriot's procedure
is indicated; in the fifth column is Descartes' exponential notation.
214
A HISTORY OF MATHEMATICAL NOTATIONS
In his Arithmetica infinitorum1 he used the colon as a symbol for
for 1/aD-a2;
aggregation, as i/:a?+l for T/a2+l, \/:oD—a*:
Oughtred's notation for ratio and proportion, -fr for continued pro-
portion. As the sign for multiplication one finds in this book X and
X, both signs occurring sometimes on one and the same page (for
instance, p. 172). In a table (p. 169) he puts D for a given number:
"Verbi gratia; si numerus hac nota D designatus supponatur cognitus,
reliqui omnes etiam cognoscentur." It is in this book and in his De
6l
*=?
} — . j 2
4*7
3 — 5
FIG. 97. — From Braiicker's translation of Halm (1668). The same arrange-
ment of the solution as in 1659, but the omicron-sigma takes the place of the
Archimedean spiral; the ordinal numbers in the outer column are not dotted,
while the number in that column which does not refer to steps in the middle
column carries a bar, 2. Step 5 means ''line 4, multiplied by 2, gives 4ab = 2DD —
27V'
sectionibus conicis that Wallis first introduces oo for infinity. He
says (p. 70) : "Cum enim primus terminus in serie Primanorum sit 0,
primus terminus in serie reciproca erit oo vel infinitus : (sicut, in
divisione, si diviso sit 0, quotiens erit infinitus)"; on pages 152, 153:
" . . . . quippe ^- (pars infinite parva) habenda erit pro nihilo,"
"oo X-&-B = B" "Nam oo, oo +1 oo — 1, perinde sunt"; on page 168:
"Quamvis enim oo XO non aliquem determinate numerum designet.
. . . ." An imitation of Oughtred is Wallis7 "HT:1|-|," which occurs in
4
his famous determination by interpolation of as the ratio of two in-
4
finite products. At this place he represents our by the symbol D.
7T
1 Johannis Wallisii Arithmetica infinitorum (Oxford, 1655).
INDIVIDUAL WRITERS 215
He says also (p. 175) : "Si igitur ut j/ : 3 X 6 : significat terminum medi-
um inter 3 et 6 in progressione Geometrica aequabili 3, 6, 12, etc.
(continue multiplicando 3X2X2 etc.) ita ]7T : 1 1| : significet terminum
medium inter 1 et f in progressione Geometrica decrescente 1, f, *£-,
etc. (continue multiplicando iXfXf, etc.) erit D=)?r:l|f: Et
propterea circulus est ad quadratum diametri, ut 1 ad »r:l|f." He
uses this symbol again in his Treatise of Algebra (1685), pages 296, 362.
7* Dt 'Witifiwif Al&dticA* ' CAP. ri^
-' Pole ft <ts (tu
Radix 3? TL ^ A * a r
Quadratutn 7^ S^ s Aq ** a* 2
Cubus <£ CT" Ac aaa a * ' 3
Q^ud. quadratum 2^ ^^ Aqq -^^4 <JA ^
Sufdcfolidtim f^ S" Aqc 8cc. tf l 5
QiiadiCubi; y£ 5P Acc tf* *
' 3Lm Surdtfolidam. B|o bS Aqqc ^^ y
Quad, quad.quad, ^ ^%%. &&!£- ^qcc 41 '8
Cubicubut <£<£ CC Accc J* 9
Quad, Surdcfol. 2^/<? ^S Aqqcc J '-* 10
3 m Surdcfol idum C/tf cS Aqccc 4"- H
Qiiad. quad. cnbi( 2^7j^« <^^p Acccc .*'•*• 12
4m Surdrfolidum D/'« d S Aqqccc rf i$ 13
Qtiad. 2* Surdcfol. - %B[# ^J3^ Aqcccc <i >A i.j,
Cubui Surdcfol. *pf^. CS Accccc a ^ i<?
Quad. quad quad. quad.
FIG. 98. — From John Wallis, Operum mathematicorum pars prima (Oxford,
1657), p. 72.
The absence of a special sign for division shows itself in such pas-
sages as (p. 135): "Ratio rationis hujus ^-~ ad illam £, puta
/LJ
- )- — (— ,erit " He uses Oughtred's clumsy notation for decimal
^/ 2CU \LH
fractions, even though Napier had used the point or comma in 1617.
On page 166 Wallis comes close to the modern radical notation; he
writes ' *\/*R" for l/R. Yet on that very page he uses the old designa-
tion "i/qqR" for
216 A HISTORY OF MATHEMATICAL NOTATIONS
His notation for continued fractions is shown in the following
quotation (p. 191):
"Esto igitur fractio ejusmodi - 6
continue fracta quaelibet, sic a @ - 5? e
designata, 5 ~ > e^c<>
where
The suggestion of the use of negative exponents, introduced later
by Isaac Newton, is given in the following passage (p. 74) : "Ubi
autem series directae indices habent 1, 2, 3, etc. ut quae supra seriem
Aequalium tot gradibus ascendunt; habebunt hae quidem (illis re-
ciprocae) suos indices contrarios negativos — 1, —2, —3, etc. tanquam
tot gradibus infra seriem Aequalium descendentes."
In Wallis' Mathesis universalis,1 the idea of positive and negative
integral exponents is brought out in the explanation of the Hindu-
Arabic notation. The same principle prevails in the sexagesimal nota-
tion, "hoc est, minuta prima, secunda, tertia, etc. ad dextram de-
scendendo," while ascending on the left are units "quae vocantur
Sexagena prima, secunda, tertia, etc. hoc modo.
\\\\ \\\ \\ \ o / // /// ////
49, 36, 25, 15, 1, 15, 25, 36, 49. "
That the consideration of sexagesimal integers of denominations of
higher orders was still in vogue is somewhat surprising.
On page 157 he explains both the "scratch method" of dividing
one number by another and the method of long division now current,
except that, in the latter method, he writes the divisor underneath
the dividend. On page 240: "A, M, V jf " for arithmetic proportion,
i.e., to indicate M — A = V—M. On page 292, he introduces a general
root d in this manner: "\/dRd = R." Page 335 contains the following
interesting combination of symbols:
/ • s * " , — * — * In Modern Symbols
"Si A • B • C : a • 0 • 7 If A:B = a:p,
' — d±_:: ^Tr-" and #:C = /3:7,
Erit A • C :: a • 7." thenA:C = ai7.
196. In the Treatise of Algebra? (p. 46), Wallis uses the decimal
point., placed at the lower terminus of the letters, thus: 3.14159,
1 Johannis Wallisii Mathe&ia universalis: sive, Arithmeticum opus integrum
(Oxford, 1657), p. 65-68.
2 Op. oil. (London, 1685).
INDIVIDUAL WRITERS 217
26535 ..... , but on page 232 he uses the comma, "12,756," ",3936."
On page 67, describing Oughtred's Clavis maihematicaey Wallis says:
"He doth also (to very great advantage) make use of several Ligatures,
or Compendious Notes, to signify the Summs, Differences, and Rec-
tangles of several Quantities. As for instance, Of two quantities A
(the Greater, and E (the Lesser,) the Sum he calls Z, the Difference
X, the Rectangle M ..... " On page 109 Wallis summarizes various
practices: "The Root of such Binomial or Residual is called a Root
universal; and thus marked \/uy (Root universal,) or j/6, (Root of a
Binomial,) or j/r, (Root of a Residual,) or drawing a Line over the
whole Compound quantity; or including it (as Oughtred used to do)
within two colons; or by some other distinction, whereby it may ap-
pear, that the note of Radicality respects, not only the single quantity
next adjoining, but the whole Aggregate. As j/6 : 2+1/3-j/r : 2—
On page 227 Wallis uses Rahn's sign -f- for division; along with the
colon as the sign of aggregation it gives rise to oddities in notation
like the following: "ll-2laa+a*: + bb."
On page 260, in a geometric problem, he writes "QAE" for the
square of the line AE; he uses fp for the absolute value of the
difference.
On page 317 his notation for infinite products and infinite series is
as follows:
etc."
etc."
" for V2-
on page 322:
On page 332 he uses fractional exponents (Newton having intro-
duced the modern notation for negative and fractional exponents in
1676) as follows:
V5:c5+c4:r-z5: or
The difficulties experienced by the typesetter in printing fractional
exponents are exhibited on page 346, where we find, for example,
"d\ x%" for d*x*. On page 123, the factoring of 5940 is shown as
follows:
"11)5)3)3)3)2)2) 5940 (2970(1485(495(165(55(11(1 ."
218 A HISTORY OF MATHEMATICAL NOTATIONS
In a letter to John Collins, Wallis expresses himself on the sign of
multiplication: "In printing my things, I had rather you make use of
Mr. Oughtred's note of multiplication, X, than that of $£; the other
being the more simple. And if it be thought apt to be mistaken for X,
it may [be] helped by making the upper and lower angles more obtuse
ixj."1 "I do not understand why the sign of multiplication X should
more trouble the convenient placing of the fractions than the other
signs + - = > ::."2
Wallis, in presenting the history of algebra, stressed the work of
Harriot and Oughtred. John Collins took some exception to Wallis'
attitude, as is shown in the following illuminating letter. Collins says:3
"You do not like those words of Vieta in his theorems, ex adjunctione
piano solidi, plus quadrato quadrati, etc., and think Mr. Oughtred
the first that abridged those expressions by symbols; but I dissent,
and tell you 'twas done before by Cataldus, Geysius, and Camillus
Gloriosus,4 who in his first decade of exercises, (not the first tract,)
printed at Naples in 1627, which was four years before the first edition
of the Clavis, proposeth this equation just as I here give it you, viz.,
lccc+ IQqcc+llqqc- 2304cc- 18364gc - 133000^ - 54505c + 3728q +
8064^ aequatur 4608, finds N or a root of it to be 24, and composeth
the whole out of it for proof, just in Mr. Oughtred's symbols and
method. Cataldus on Vieta came out fifteen years before, and I can-
not quote that, as not having it by me And as for Mr. Ought-
red's method of symbols, this I say to it; it may be proper for you as a
commentator to follow it, but divers I know, men of inferior rank that
have good skill in algebra, that neither use nor approve it Is
not Ab sooner wrote than Aqcf Let A be 2, the cube of 2 is 8, which
squared is 64: one of the questions between Magnet Grisio and
Gloriosus is whether 64 = Acc or Aqc. The Cartesian method tells you
it is A6j and decides the doubt."
EXTRACT FROM ACTA ERUDITORUM5
197. "Monendurn denique, nos in posterum in his Actis usuros esse
Signis Leibnitianis, ubi cum Algebraicis res nobis fuerit, ne typothetis
1 John Wallis to John Collins, July 21, 1668 (S. P. Rigaud, Correspondence
of Scientific Men of the Seventeenth Century, Vol. II [Oxford, 1841], p. 492).
2 Wallis to Collins, September 8, 1668 (ibid., p. 494).
3 Letter to John Wallis, about 1667 (ibid., p. 477-80).
4 " fixer citationum Mathematicarum Decas prima, Nap. 1627, and probably
Cataldus' Transformatio Geometrica, Bonon. 1612."
5 Taken from Ada eruditorum (Leipzig, 1708), p. 271.
INDIVIDUAL WRITERS 219
tacdia & molestias gratis creemus, utque ambiguitatcs evitemus.
Loco igitur lineolae characteribus supraducendae parenthcsin ad-
hibebimus, imrno in multiplicatione simplex comma, ex. gr. loco
Vaa+bb scribemus V(aa+bb) & pro aa+bbXc ponemus aa+bb, c.
Divisionem designabimus per duo puncta, nisi peculiaris quacdam
circumstantia morem vulgarem adhiberi suaserit. Ita nobis erit
a:& = £. Et hinc peculiaribus signis ad denotandam proportionem
nobis non erit opus. Si enim f uerit ut a ad 6 ita c ad d, erit a:b — c:d.
Quod potentias attinet, aa+bb™ designabimus per (aa+bb)m: unde
& Vaa+bb erit— (aa+66)1 : m & Vaa+bbn=(aa+bb)n:m. Nulli vero
dubitamus fore, ut Geometrae omnes Acta haec legentes Signorum
Leibnitianorum praestantiam animadvertant, & nobiscum in eadem
consentiant."
The translation is as follows: "We hereby issue the reminder that
in the future we shall use in these Acta the Leibnizian signs, where,
when algebraic matters concern us, we do not choose the typographi-
cally troublesome and unnecessarily repugnant, and that we avoid
ambiguity. Hence we shall prefer the parenthesis to the characters
consisting of lines drawn above, and in multiplication by all means
simply the comma; for example, in place of I/ aa+bb we write
l/(aa+W) and for aa+bbXc we take aa+bbt c. Division we mark
with two dots, unless indeed some peculiar circumstance directs ad-
herence to the usual practice. Accordingly, we have a:6 = r. And it
is not necessary to denote proportion by any special sign. For, if a
is to b as c is to d, we have a:b = c:d. As regards powers, aa+bbm,
we designate them by (aa+bb)m; whence also V aa+bb becomes
m/—~ "
= (aa+W>)1:mand ^aa+bbn = (aa+bb)n:m. We do not doubt that all
geometers who read the Acta will recognize the excellence of the
Leibnizian symbols and will agree with us in this matter."
EXTRACT FROM MISCELLANEA BEROLINENSIA1
198. "Monitum De Characteribus Algebraicis. — Quoniam variant
Geometrae in characterum usu, nova praesertim Analysi inventa;
quae res legentibus non admodum provectis obscuritatem parit;
ideo e re visum est exponere, quomodo Characteres adhibeantur
Leibnitiano more, quern in his Miscellaneis secuturi sumus. Literae
1 Taken from Miscellanea Berolinensia (1710), p. 155. Article due to G. W,
Leibniz.
220 A HISTORY OF MATHEMATICAL NOTATIONS
minusculae o, 6, x, y solent significare magnitudines, vel quod idem
est, numeros indeterminatos: Majusculae verb, ut A, B, X, Y puncta
figurarum; ita ab significat factum ex a in 6, sed AB rectam & puncto A
ad punctum B ductam. Huic tamen observationi adeo alligati non
sumus, ut non aliquando minusculas pro punctis, majusculas pro
numeris vel magnitudinibus usurpemus, quod facile apparebit ex
modo adhibendi. Solent etiam literae priores, ut a, 6, pro quantitati-
bus cognitis vel saltern determinatis adhiberi, sed posteriorcs, ut
x, y, pro incognitis vel saltern pro variantibus.
"Interdum pro literis adhibentur Numeri, sed qui idem significant
quod literae, utiliter tamen usurpantur relationis exprimendae gratia.
Exempli causa: Sint binae aequationes generales secundi gradus pro
incognita, x; eas sic exprimere licebit: 10xx> — }-*llx> — £-12 = 0 &
2Qxx> — {— 2l£> — 1—22 = 0 ita in progressu calculi ex ipsa notatione
apparet quantitatis cujusque relatio; nempe 21 (ex. gr.) per notam
dextram, quae est 1 agnoscitur esse coefficiens ipsius x simplicis, at
per notam sinistram 2 agnoscitur esse ex. aeq. secunda: sed et servatur
lex quaedam homogeneorum. Et ope harum duarum aequationum
tollendo x, prodit aequatio, in qua similiter sc haberc oportet 10, 11,
12 et 12, 11, 10; item 20, 21, 22 et 22, 21, 20; et dcniquc 10, 11, 12 se
habent ut. 20, 21, 22. id est si pro 10, 11, 12 substituas 20, 21, 22 et
vice versa manet eadem aequatio; idemque est in caeteris. Tales
numeri tractantur ut literae, veri autern numeri, discriminis causa,
parenthesibus includuntur vel aliter discernuntur. Ita in tali sensu
11.20. significat numeros indefinites 11 et 20 in se invicem ductos, non
vero significat 220 quasi esscnt Numeri veri. Sed hie usus ordinarius
non est, rariusque adhibetur.
"Signa, Additionis nimirum et Subtractions , sunt > — I- plus, —
minus, > — J- plus vel minus, > j . priori oppositum minus vel plus. At
( — H) vel ( ... | . ) est nota ambiguitatis signorum, independens a
priori; et (( — {-) vel (( j .) alia independens ab utraque; Differt
•autern Signum ambiguum a Differentia quantitatum, quae etsi aliquan-
do incerta, non tamen ambigua est Sed differentia inter a et
by significat a — fe, si a sit majus, et 6 — a si b sit majus, quod etiam ap-
pellari potest moles ipsius a— 6, intelligendo (exempli causa) ipsius
— 1-2 et ipsius — 2 molem esse eandem, nempe — £-2; ita si a— 6
vocemus c utique mol. c, seu moles ipsius c erit — 1-2, quae est quan-
titas affirmativa sive c sit affirmativa sive negativa, id est, sive sit c
idem quod — I- 2, sive c sit idem quod 2. Et quantitates duae
diversae eandem molem habentes semper habent idem quadratum.
INDIVIDUAL WRITERS 221
"Multiplicationem plerumque signifare content! sumus per nudam
appositionem: sic ab significat a multiplicari per 6, Numeros multi-
plicantes solemus praefigere, sic 3a significat triplum ipsius a interdum
tamen punctum vel comma interponimus inter multiplicans et
multiplicandum, velut cum 3, 2 significat 3 multiplicari per 2, quod
facit 6, si 3 et 2 sunt nurneri veri; et AB, CD significat rectam AB
duci in rectam CD, atque inde fieri rectangulum. Sed et commata inter-
dum hoc loco adhibemus utiliter, velut a, b*%*c, vel AB, CD — k~EF, id
est, a duci in 6 — f-c, vel AB in CD — }-EF; sed de his mox, ubi de
vinculis. Porro propria Nota Multiplicationis non solet esse neces-
saria, cum plerumque appositio, qualem diximus, sufficiat. Si tamen
utilis aliquando sit, adhibebitur potius r\ quam M , quia hoc ambigui-
tatem parit, et ita AB/^CD significat AB duci in CD.
"Diviso significatur interdum more vulgari per subscriptionem
diuisoris sub ipso dividendo, intercedente linea, ita a dividi per 6,
significatur vulgo per 7 ; plerumque tamen hoc evitare praestat,
efficereque, ut in eadem linea permaneatur, quod sit interpositis
duobus punctis; ita ut a: 6 significat a dividi per 6. Quod si a: b rursus
dividi debeat per c, poterimus scribere a : b, : c, vel (a : &) : c. Etsi enim
res hoc casu (sane simplici) facile aliter exprimi posset, fit enim
a : (be) vel a : be non tamen semper divisio actu ipse f acienda cst, sed
saepe tantum indicanda, et tune praestat operationis dilatae pro-
cessum per commata vel parentheses indicari ..... Et exponens inter-
dum lineolis includitur hac modo (T](AjB — l-BC) quo significatur
cubus rectac AB - — ^BC ..... a€+n et utiliter interdum lineola sub-
ducitur, ne literae exponentiales aliis confundantur; posset etiam
scribi fe+n^ a .....
" ____ itav/(a3) vel i/[j[](a3) rursus est a, .... sed f/Z vel i/02
significat radicern cubicam ex eodem numero, et -fr'2 vel i/ Q 2 signifi-
cat, radicem indeterminati gradus e ex 2 extrahendam .....
"Pro vinculis vulgo solent adhiberi ductus linearum; sed quia
lineis una super alia ductis, saepe nimium spatii occupatur, aliasque
ob causas commodius plerumque adhibentur commata et parentheses.
Sic a, b+f^c idem est quod a, 6*4 — <c vel a(6-J — <c); et a-\ — <b,
idem quod a-f — <b, c*-J — <d vel (a -4 — <) (c-J — <), id est,
multiplicatum per c«~{ — «d. Et similiter vincula in vin-
culis exhibentur. Ita a, 6c»-{ — <e/H — <g etiam sic exprimetur,
Et a, be -4 — ce/-J — <g+i — <Wm, n potest etiam
222 A HISTORY OF MATHEMATICAL NOTATIONS
sic exprimi: »-J — <(a(bc> — l^e(f+g))+hl?n)n. Quod de vinculis multi-
plicationis, idem intelligi potcst de vinculis divisionis, exempli gratia
b T e * I . .. , . r
- -I — < 7-1 , — sic scnbetur in una Imea
c
n
nihilque in his difficultatis, modo teneamus, quicquid parenthesin
aliquam implet pro una quantitate haberi, .... Idemque igitur
locum habet in vinculis cxtractionis radicalis.
Sic
\a4-! — <^ e,/*-f— <<7 idem est quod i/(a4
Et pro i/aa -I — < b|/cc H — < dd
e —J — < V'fi/gg • I c AA —I — < fcfc
scribi poterit j/ (aa -f — < 6 j/(cc -J — < dd)) : ,
itaque a = 6 significat, a, esse equale ipsi 6, et a=^6 significat a esse
majus quam 6, et a =—b significat a esse minus quarn b.
"Sed et proportionalitas vel analogia de quantitatibus enunciatur,
id est, rationis identitas, quam possumus in Calculo exprimere per
notam aequalitatis, ut non sit opus pcculiaribus notis. Itaqua a
esse ad 6, sic ut I ad m, sic exprimere poterimus a:b = l: m, id est y = .
Nota continue proportionalium erit -H-, ita ut -H- a.6.c. etc. sint con-
tinue proportionales. Interdum nota Similitudinis prodest, quae est
c^ , item nota similitudinis aequalitatis simul, seu nota congruitatis & 9
Sic DEF v> PQR significant Triangula haec duo esse similia; at DEF &
PQR significant congruere inter se. Huic si tria inter se habeant
eandem rationem quam tria alia inter se, poterimus hoc exprimere
nota similitudinis, ut a; 6; a> Z; w; n quod significat esse a ad 6, ut I ad
m, et a ad c ut I ad n, et b ad c ut m ad n ..... "
The translation is as follows:
"Recommendations on algebraic characters. — Since geometers differ
in the use of characters, especially those of the newly invented anal-
ysis, a situation which perplexes those followers who as yet are not
very far advanced, it seems proper to explain the manner of using the
characters in the Leibnizian procedure, which we have adopted in the
INDIVIDUAL WRITERS 223
Miscellanies. The small letters a, &, x, y, signify magnitudes, or what
is the same thing, indeterminate numbers. The capitals on the other
hand, as A, 5, X, F, stand for points of figures. Thus ab signifies the
result of a times 6, but AB signifies the right line drawn from the point
A to the point B. We are, however, not bound to this convention, for
not infrequently we shall employ small letters for points, capitals for
numbers or magnitudes, as will be easily evident from the mode of
statement. It is customary, however, to employ the first letters a, 6,
for known or fixed quantities, and the last letters x, y, for the un-
knowns or variables.
"Sometimes numbers are introduced instead of letters, but they
signify the same as letters; they are convenient for the expression of
relations. For example, let there be two general equations of the
second degree having the unknown x. It is allowable to express them
thus: Hte+ llz+ 12 = 0 and 2(te+21z+22 = 0. Then, in the prog-
ress of the calculation the relation of any quantity appears from the
notation itself; thus, for example, in 21 the right digit which is 1
is recognized as the coefficient of x, and the left digit 2 is recognized
as belonging to the second equation; but also a certain law of homo-
geneity is obeyed. And eliminating x by means of these two equa-
tions, an equation is obtained in which one has similarity in 10, 11, 12
and 12, 11, 10; also in 20, 21, 22 and 22, 21, 20; and lastly in 10, 11, 12
and 20, 21, 22. That is, if for 10, 11, 12, you substitute 20, 21, 22 and
vice versa, there remains the same equation, and so on. Such numbers
are treated as if letters. But for the sake of distinction, they are in-
cluded in parentheses or otherwise marked. Accordingly, 11-20.
signifies the indefinite numbers 11 and 20 multiplied one into the
other; it does not signify 220 as it would if they were really numbers.
But this usage is uncommon and is rarely applied.
"The signs of addition and subtraction are commonly + plus,
— minus, ± plus or minus, T the opposite to the preceding, minus
or plus. Moreover ( ± ) or ( + ) is the mark of ambiguity of signs that
are independent at the start; and ((±) or (( + ) are other signs inde-
pendent of both the preceding. Now the symbol of ambiguity differs
from the difference of quantities which, although sometimes unde-
termined, is not ambiguous But a — b signifies the difference
between a and b when a is the greater, b — a when 6 is the greater,
and this absolute value (moles) may however be called itself a — 6, by
understanding that the absolute value of +2 and —2, for example, is
the same, namely, +2. Accordingly, if a— b is called c, thenraoZ. c or
the absolute value of c is +2, which is an affirmative quantity whether
224 A HISTORY OF MATHEMATICAL NOTATIONS
c itself is positive or negative; i.e., either c is the same as +2, or c is
the same as —2. Two different quantities having the same absolute
value have always the same square.
"Multiplication we are commonly content to indicate by simple
apposition: thus, ab signifies a multiplied by 6. The multiplier we are
accustomed to place in front; thus 3a means the triple of a itself.
Sometimes, however, we insert a point or a comma between multi-
plier and multiplicand; thus, for example, 3,2 signifies that 3 is multi-
plied by 2, which makes 6, when 3 and 2 are really numbers; and
AB,CD signifies the right line AB multiplied into the right line CD,
producing a rectangle. But we also apply the comma advantageously
in such a case, for example,1 as a,6+c, or AB,CD+EF; i.e., a multi-
plied into 6+c, or AB into CD+EF; we speak about this soon, under
vinculums. Formerly no sign of multiplication was considered neces-
sary for, as stated above, commonly mere apposition sufficed. If,
however, at any time a sign seems desirable use r^ rather than ><! ,
because the latter leads to ambiguity; accordingly, AB^\CD sig-
nifies AB times CD.
"Division is commonly marked by writing the divisor beneath its
dividend, with a line of separation between them. Thus a divided by
b is ordinarily indicated by r ; often, however, it is preferable to avoid
this notation and to arrange the signs so that they are brought into
one and the same line; this may be done by the interposition of two
points; thus a:b signifies a divided by b. If a:b in turn is to be divided
by c, we may write a : 6, : c, or (a : b) : c. However, this should be ex-
pressed more simply in another way, namely, a : (be) or a : be, for the
division cannot always be actually carried out, but can be only
indicated, and then it becomes necessary to mark the delayed process
of the operation by commas or parentheses Exponents are
frequently inclosed by lines in this manner [a] (AB-\-BC), which
means the cube of the line AB+BC . . . . ; the exponents of al+n
may also be advantageously written between the lines, so that the
literal exponents will not be confounded with other letters; thus it-
may be written \l+n\ a From ^(a3) or j/ B (a3) arises a . . . . ;
but 1^2 or i/E 2 means the cube root of the same number, and •$/%
or V 02 signifies the extraction of a root of the indeterminate
order e
"For aggregation it is customary to resort to the drawing of
1 A similar use of the comma to separate factors and at the same time express
aggregation occurs earlier in He*rigone (see § 189).
INDIVIDUAL WRITERS 225
lines, but because lines drawn one above others often occupy too
much space, and for other reasons, it is often more convenient to
introduce commas and parentheses. Thus a, b+c is the same as
a, b+c or a(6+c) ; and a+b, c+d is the same as a+b, c+d, or (a+b)
(c+d), i.e., +a+b multiplied by c+d. And, similarly, vinculums are
placed under vinculums. For example, a, bc+ef+g is expressed also
thus, a(bc+e(f+g)), and a, bc+ef+g+hlm,n may be written also
+ (a(bc+e(f+g))+hlm)n. What relates to vinculums in multiplica-
tion applies to vinculums in division. For example,
*+_!_ J.
— - — may be written in one line thus:
and there is no difficulty in this, as long as we observe that whatever
fills up a given parenthesis be taken as one quantity ..... The same
is true of vinculums in the extraction of roots. Thus ^a4+l/e, f+g
is the same as ]/(a4+i/(e(/+fir))) or \/(a4+^/(e, f+g)). And
- ., /f . , ,f , ,,.N ,
for - -- - ...... — — — — one may write y(aa+oy(cc+da)):9e+
Again a = b signifies that a is equal to 6, and
a=~b signifies that a is greater than 6, and a=-b that a is less than b.
Also proportionality or analogia of quantities, i.e., the identity of ratio,
may be represented; we may express it in the calculus by the sign of
equality, for there is no need of a special sign. Thus, we may indi-
cate that a is to b as I is to ra by a:b = l:m, i.e., r =— . The sign for
o iii
continued proportion is -Jf , so that TT a, b, c, and d are continued pro-
portionals.
"There is adopted a sign for similitude; it is <*> ; also a sign for
both similitude and equality, or a sign of congruence, & accordingly,
DEF&PQR signifies that the two triangles are similar; but DEF&.
PRQ marks their congruence. Hence, if three quantities have to one
another the same ratio that three others have to one another, we may
mark this by a sign of similitude, asa;6;ccoZ;m;n means that a is to
b as I is to m, and a is to c as / is to n, and b is to c as m is to n ..... "
In the second edition of the Miscellanea Berolinensia, of the year
226 A HISTORY OF MATHEMATICAL NOTATIONS
1749, the typographical work is less faulty than in the first edition of
1710; some slight errors are corrected, but otherwise no alterations
are made, except that Harriot's signs for "greater than" and "less
than" are adopted in 1749 in place of the two horizontal lines of un-
equal length and thickness, given in 1710, as shown above.
199. Conclusions. — In a letter to Collins, John Wallis refers to a
change in algebraic notation that occurred in England during his
lifetime : "It is true, that as in other things so in mathematics, fashions
will daily alter, and that which Mr. Oughtred designed by great
letters may be now by others be designed by small; but a mathemati-
cian will, with the same ease and advantage, understand Ac, and a3
or aaa."1 This particular diversity is only a trifle as compared with
what is shown in a general survey of algebra in Europe during the
fifteenth, sixteenth, and seventeenth centuries. It is discouraging to
behold the extreme slowness of the process of unification.
In the latter part of the fifteenth century p and m became symbols
for "plus" and "minus" in France (§ 131) and Italy (§ 134). In Ger-
many the Greek cross and the dash were introduced (§ 146). The two
rival notations competed against each other on European territory
for many years. The p and m never acquired a foothold in Germany.
The German + and — gradually penetrated different parts of Europe.
It is found in Scheubel's Algebra (§ 158), in Recorders Whetstone of
Witte, and in the Algebra of Clavius. In Spain the German signs occur
in a book of 1552 (§ 204), only to be superseded by the p and m in
later algebras of the sixteenth century. The struggle lasted about
one hundred and thirty years, when the German signs won out every-
where except in Spain. Organized effort, in a few years, could have
ended this more than a century competition.
If one takes a cross-section of the notations for radical expressions
as they existed in algebra at the close of the sixteenth century, one
finds four fundamental symbols for indicating roots, the letters B and Z,
the radical sign j/ proper and the fractional exponent. The letters
8 and / were sometimes used as capitals and sometimes as small
letters (§§ 135, 318-22). The student had to watch his step, for at times
these letters were used to mark, not roots, but the unknown quantity
r and, perhaps, also its powers (§ 136). When & stood for "root," it
became necessary to show whether the root of one term or of several
terms was meant. There sprang up at least seven different symbols
For the aggregation of terms affected by the ft, namely, one of Chuquet
(§ 130), one of Pacioli (§ 135), two of Cardan (§ 141), the round paren-
1 See Rigaud, op. dt.t Vol. II, p. 475.
INDIVIDUAL WRITERS 227
thesis of Tartaglia (§ 351), the upright and inverted letter L of Bombelli
(§ 144), and the r bin. and r trinomia of A. V. Roomen (§ 343). There
were at least five ways of marking the orders of the root, those of
Chuquet (§ 130), De la Roche (§ 132), Pacioli (§ 135), Ghaligai
(§ 139), and Cardan (Fig. 46). With A. M. Visconti1 the signs
R.ce cu. meant the "sixth root"; he used the multiplicative principle,
while Pacioli used the additive one in the notation of radicals. Thus
the letter # carried with it at least fifteen varieties of usage. In con-
nection with the letter /, signifying latus or "root/' there were at least
four ways of designating the orders of the roots and the aggregation
of terms affected (§§ 291, 322). A unique line symbolism for roots of
different orders occurs in the manuscripts of John Napier (§ 323).
The radical signs for cube and fourth root had quite different
shapes as used by Rudolff (§§ 148, 326) and Stifel (§ 153). Though
clumsier than StifePs, the signs of Rudolff retained their place in some
books for over a century (§ 328). To designate the order of the roots,
Stifel placed immediately after the radical sign the German abbrevia-
tions of the words zensus, cubus, zensizensuSj sursolidus, etc. Stevin
(§ 163) made the important innovation of numeral indices. He placed
them within a circle. Thus he marked cube root by a radical sign
followed by the numeral 3 coraled in a circle. To mark the root of an
aggregation of terms, Rudolff (§§ 148, 348) introduced the dot placed
after the radical sign ; Stifel sometimes used two dots, one before the
expression, the other after. Stevin (§§ 163, 343) and Digges (§§ 334,
343) had still different designations. Thus the radical sign carried
with it seven somewhat different styles of representation. Stevin
suggested also the possibility of fractional exponents (§ 163), the
fraction being placed inside a circle and before the radicand.
Altogether there were at the close of the sixteenth century twenty-
five or more varieties of symbols for the calculus of radicals with which
the student had to be familiar, if he desired to survey the publications
of his time.
Lambert Lincoln Jackson makes the following historical observa-
tions: "For a hundred years after the first printed arithmetic many
writers began their works with the line-reckoning and the Roman
numerals, and followed these by the Hindu arithmetic. The teaching
of numeration was a formidable task, since the new notation was so
unfamiliar to people generally/'2 In another place (p. 205) Jackson
1 "Abbreviationes," Praclica numerorum, ct mcnsurarum (Brescia, 1581).
2 The Educational Significance of Sixteenth Century Arithmetic (New York,
1906), p. 37, 38.
228 A HISTORY OF MATHEMATICAL NOTATIONS
states: "Any phase of the growth of mathematical notation is an
interesting study, but the chief educational lesson to be derived is that
notation always grows too slowly. Older and inferior forms possess
remarkable longevity, and the newer and superior forms appear feeble
and backward. We have noted the state of transition in the sixteenth
century from the Roman to the Hindu system of characters, the intro-
duction of the symbols of operation, +, — , and the slow growth
toward the decimal notation. The moral which this points for
twentieth-century teachers is that they should not encourage history
to repeat itself, but should assist in hastening new improvements."
The historian Tropfke expresses himself as follows: "How often
has the question been put, what further achievements the patriarchs
of Greek mathematics would have recorded, had they been in posses-
sion of our notation of numbers and symbols! Nothing stirs the his-
torian as much as the contemplation of the gradual development of
devices which the human mind has thought out, that he might ap-
proach the truth, enthroned in inaccessible sublimity and in its fullness
always hidden from earth. Slowly, only very slowly, have these de-
vices become what they are to man today. Numberless strokes of the
file were necessary, many a chink, appearing suddenly, had to be
mended, before the mathematician had at hand the sharp tool with
which he could make a successful attack upon the problems con-
fronting him. The history of algebraic language and writing presents
no uniform picture. An assemblage of conscious and unconscious
innovations, it too stands subject to the great world-law regulating
living things, the principle of selection. Practical innovations make
themselves felt, unsuitable ones sink into oblivion after a time. The
force of habit is the greatest opponent of progress. How obstinate
was the struggle, before the decimal division met with acceptation,
before the proportional device was displaced by the equation, before
the Indian numerals, the literal coefficients of Vieta, could initiate a
world mathematics."1
Another phase is touched by Treutlein: "Nowhere more than in
mathematics is intellectual content so intimately associated with the
form in which it is presented, so that an improvement in the latter
may well result in an improvement of the former. Particularly in
arithmetic, a generalization and deepening of concept became pos-
sible only after the form of presentation had been altered. The his-
tory of our science supplies many examples in proof of this. If the
Greeks had been in possession of our numeral notation, would their
1 Tropfke, GeschichtederElementar-Mathematik, Vol. II (Leipzig, 1921), p. 4, 5.
ADDITION AND SUBTRACTION 229
mathematics not present a different appearance? Would the binomial
theorem have been possible without the generalized notation of pow-
ers? Indeed could the mathematics of the last three hundred years
have assumed its degree of generality without Vieta's pervasive
change of notation, without his introduction of general numbers?
These instances, to which others from the history of modern mathe-
matics could be added, show clearly the most intimate relation between
substance and form."1
B. SPECIAL SURVEY OF THE USE OF NOTATIONS
SIGNS FOR ADDITION AND SUBTRACTION
200. Early symbols. — According to Hilprecht,2 the early Baby-
lonians had an ideogram, which he transliterates LAL, to signify
"minus." In the hieratic papyrus of Ahines and, more clearly in the
hieroglyphic translation of it, a pair of legs walking forward is the
sign of addition; away, the sign of subtraction.3 In another Egyptian
papyrus kept in the Museum of Fine Arts in Moscow,4 a pair of legs
walking forward has a different significance; there it means to square
a number.
Figure 99, translated, is as follows (reading the figure from right
to left) :
"5 added and J- [of this sum] taken away, 10 remains.
Make -fV of this 10: the result is 1, the remainder 9.
| of it, namely, 6, added to it; the total is 15. £ of it is 5.
When 5 is taken away, the remainder is 10."
In the writing of unit fractions, juxtaposition meant addition, the
unit fraction of greatest value being written first and the others in
descending order of magnitude.
While in Diophantus addition was expressed merely by juxtaposi-
tion (§ 102), a sporadic use of a slanting line / for addition, also a
semi-elliptical curve 7 for subtraction, and a combination of the two
1 Treutlein, "Die deutsche Coss," Abhandlungen z. Geschichte der Mathematik,
Vol. II (Leipzig, 1879), p. 27, 28.
2 H. V. Hilprecht, Babylonian Expedition: Mathematical etc. Tablets (Phila-
delphia, 1906), p. 23.
3 A. Eisenlohr, op. cit. (2d ed.), p. 46 (No. 28), 47, 237. See also the improved
edition of the Ahmes papyrus, The Rhind Mathematical Papyrus , by T. Eric Feet
(London, 1923), Plate J, No. 28; also p. 63.
4 Peet, op. cit., p. 20, 135: Ancient Egypt (1917), p. 101.
230 A HISTORY OF MATHEMATICAL NOTATIONS
P for the total result has been detected in Greek papyri.1 Diophantus'
sign for subtraction is well known (§ 103). The Hindus had no mark
for addition (§ 106) except that, in the Bakhshali Arithmetic, yu is
used for this purpose (§ 109). The Hindus distinguished negative
quantities by a dot (§§ 106, 108), but the Bakhshali Arithmetic uses
the sign + for subtraction (§ 109). The Arab al-Qalasadi in the fif-
teenth century indicated addition by juxtaposition and had a special
sign for subtraction (§ 124). The Frenchman Chuquet (1484), the
Italian Pacioli (1494), and the sixteenth-century mathematicians in
Italy used p or p: for plus and m or m: for "minus" (§§ 129, 134).
\ * ®
FIG. 99. — From the hieroglyphic translation of the Ahmes papyrus, Problem
28, showing a pair of legs walking forward, to indicate addition, and legs walking
away, to indicate subtraction. (Taken from T. E. Peet, The Rhind Mathematical
Papyrus, Plate J, No. 28.)
201. Origin and meanings of the signs + and — . — The modern
algebraic signs + and — came into use in Germany during the last
twenty years of the fifteenth century. They are first found in manu-
scripts. In the Dresden Library there is a volume of manuscripts,
C. 80. One of these manuscripts is an algebra in German, written in
the year 148 1,2 in which the minus sign makes its first appearance in
1H. Brugsch, Numerorum apud veteres Aegyptios demoticorum doctrina. Ex
papyris (Berlin, 1849), p. 31; see also G. Friedlein, Zahlzeichen und das elementare
Rechnen (Erlangen, 1869), p. 19 and Plate I.
2 E. Wappler, Abhandlungen zur Geschichte der Mathematik, Vol. IX (1899), p.
539, n. 2; Wappler, Zur Geschichte der deutschen Algebra im 15. Jahrhundert, Zwick-
auer Gymnasialprogramm von 1887, p. 11-30 (quoted by Cantor, op. cit.t Vol. II [2d
ed., 1900], p. 243, and by Tropfke, op. cit.t Vol. II [2d ed., 1921], p. 13).
ADDITION AND SUBTRACTION 231
algebra (Fig. 100); it is called minnes. Sometimes the — is placed
after the term affected. In one case —4 is designated "4 das ist — ."
Addition is expressed by the word vnd.
In a Latin manuscript in the same collection of manuscripts,
C. 80, in the Dresden Library, appear both symbols + and — as
signs of operation (Fig. 101), but in some rare cases the + takes the
place of et where the word does not mean addition but the general
"and."1 Repeatedly, however, is the word et used for addition.
It is of no little interest
that J. Widman, who first used 1. llteete* Minuaieiclien.
the + and - in print, studied D*esd. C. 80. Deutsche Algebra, fol. 8G8'
xl. J. ' X • xl (Um 1486)
these two manuscripts in the
manuscript volume C. 80 of t*\ &f — zz
the Dresden Library and, in
fact, annotated them. One of
the Dresden Library and, in
his marginal notes is shown in ^ Fl°- 1™'^A'm™ slgl? in a r
-n- fno wj i 4. i MS» c- 80> Dresden Library. (Taken
Figure 102. Widman lectured ^ j Tro'pfke? op ^ VoLY[1921],
at the University of Leipzig, p< 14,)
and a manuscript of notes
taken in 1486 by a pupil is preserved in the Leipzig Library (Codex
Lips. 1470) .2 These notes show a marked resemblance to the two
Dresden manuscripts.
The view that our + sign descended from one of the florescent
forms for et in Latin manuscripts finds further support from works on
2. iltestes Pluazeichen.
Dread. C. 80. Lat Algebra, foL 350' 4- Dread- C- 8()-
(urn 1486) Lateinische Algebra, fol. 852*
«* + 2s* 10-*
FIG. 101. — Plus and minus signs in a Latin MS, C. 80, Dresden Library.
(Taken from Tropfke, op. cit., Vol. II [2d ed., 1921], p. 14.)
paleography. J. L. Walther3 enumerates one hundred and two differ-
ent abbreviations found in Latin manuscripts for the word et; one of
these, from a manuscript dated 1417, looks very much like the modern
1 Wappler, Programm (1887), p. 13, 15.
2 Wappler, Zeitschrift Math. u. Physik, Vol. XLV (Hist. lit. Abt, 1900), p. 7-9.
8 Lexicon diplomaticvm abbreviationes syllabarvm et vocvm in diplomatibvs et
codidbvs a secvlo VIII. ad XVI ..... Studio Joannis Lvdolfi VValtheri ....
(Ulmae, 1756), p. 456-59.
232 A HISTORY OF MATHEMATICAL NOTATIONS
+ . The downward stroke is not quite at right angles to the horizontal
stroke, thus -V.
Concerning the origin of the minus sign ( — ), we limit ourselves to
the quotation of a recent summary of different hypotheses: "One
knows nothing certain of the origin of the sign — ; perhaps it is a
simple bar used by merchants to separate the indication of the tare,
for a long time called minus, from that of the total weight of merchan-
dise; according to L. Rodet (Actes
Zusatz von WIDMANN. Soc. philol Alen$on, Vol. VIII [1879],
5. Dresd^C. so, fol. 349' p 105) this sign was derived from an
Egyptian hieratic sign. One has also
_3 *"C 380 x sought the origin of our sign — in the
(44^ — (T-i& 144- e x sign employed by Heron and Dio-
phantus and which changed to T be-
FIG. 102.— Widmari's margin- * ., . ^i A-n i
id note to MS C. 80, Dresden fore li became ~' Others stl11 havc
Library. (Taken fromTropfke.) advanced the view that the sign -
has its origin in the 6/3e\6s of the Alex-
andrian grammarians. None of these hypotheses is supported by
plausible proof."1
202. The sign + first occurs in print in Widman's book in the
question: "Als in diese exepel 16 elln pro 9 fl -J vn | + JL eynss fl wy
kume 36 elln machss alsso Addir -J- vn -\ vn J zu samen kumpt -jj I- eynss
fl Nu secz vn machss nach der regl vn kume 22 fl 8L0 eynsz fl dz ist
gerad 3 hlr in gold."2 In translation: "Thus in this example, 16 ells
[are bought] for 9 florins [and] J- and £+£ of a florin, what will 36 ells
cost? Proceed thus: Add £ and \ and | obtaining fj of a florin.
Now put down and proceed according to the rule and there results
22 florin, and /0 of a florin which is exactly 3 heller in gold." The +
in this passage stands for "and." Glaisher considers this + a mis-
print for vn (the contraction for vnnd, our "and"), but there are other
places in Widman where + clearly means "and," as we shall see
later. There is no need of considering this + a misprint.
On the same leaf Widman gives a problem on figs. We quote
from the 1498 edition (see also Fig. 54 from the 1526 edition) :
1 Encycloptdie des scien. math., Tome I, Vol. I (1904), p. 31, 32, n. 145.
2 Johann Widman, Behede vnd hubsche Rechenung auff alien Kauffmanschafft
(Leipzig, 1489), unnumbered p. 87. Our quotation is taken from J. W. L. Glaish-
er's article, "On the Early History of Signs -f- and — and on the Early German
Arithmeticians," Messenger of Mathematics, Vol. LI (1921-22), p. 6. Extracts
from Widman are given by De Morgan, Transactions of the Cambridge Philosophical
Society, Vol. XI, p. 205, and by Boncompagni, Bullelino, Vol. IX, p. 205.
ADDITION AND SUBTRACTION 233
"Veygen. — Itm Eyner Kaufft 13 lagel veygen vn nympt ye 1 ct
pro 4 fl % ort Vnd wigt itliche lagel als dan hye nochuolget. vn ich wolt
wissen was an der sum brecht
4+ 5 Wiltu dass
4 — 17 wyssen der
3+36 dess gleichn
4—19 Szo sum —
3+44 mir die ct
3+22 Vnd Ib vn
Czentner 3 — 11 Ib was — ist
3+50 dz ist mi9
4—16 dz secz besu
3+44 der vn wer
3+29 de 4539
3-12 lb(Sodu
3+9 die ct zcu Ib
gemacht hast Vnnd das + das ist mer dar zu addirest) vnd 75 min9
Nu solt du fur holcz abschlahn albeg fur eyn lagel 24 Ib vn dz ist 13
mol 24' vn macht 312 Ib dar zu addir dz — dz ist 75 Ib vnnd werden
387 Die subtrahir vonn 4539 Vnnd pleybn 4152 Ib Nu sprich 100 Ib
das ist 1 ct pro 4 fl | wie kummen 4152 Ib vnd kumen 171 fl 5 ss 4 hlr
| Vn ist recht gemacht. m
In free translation the problem reads: "Figs. — Also, a person buys
13 barrels of figs and receives 1 centner for 4 florins and \ ort (4J flor-
ins), and the weight of each barrel is as follows: 4 ct+5 Ib, 4 ct— 17 Ib,
3 ct+36 Ib, 4 ct- 19 Ib, 3 ct+44 Ib, 3 ct+22 Ib, 3 ct- 11 Ib, 3 ct+50
Ib, 4 ct-16 Ib, 3 ct+44 Ib, 3 ct+29 Ib, 3 ct-12 Ib, 3 ct+9 Ib; and I
would know what they cost. To know this or the like, sum the ct and
Ib and what is — , that is minus, set aside, and they become 4539 Ib
(if you bring the centners to Ib and thereto add the +, that is more)
and 75 minus. Now you must subtract for the wood 24 Ib for each
barrel and 13 times 24 is 312 to which you add the — , that is 75 Ib
and it becomes 387 which subtract from 4,539 and there remains
4152 Ib. Now say 100 Ib that is 1 ct for 4J fl, what do 4152 Ib come
to, and they come to 171 fl 5 ss 4| hlr which is right. "
Similar problems are given by Widman, relating to pepper and
soap. The examination of these passages has led to divergent opinions
on the original significance of the + and — . De Morgan suspected
1 The passage is quoted and discussed by Enestrom, Bibliotheca mathematica,
Vol. IX (3d ser., 1908-9), p. 156, 157, 248; see also ibid., Vol. VIII, p. 199.
234 A HISTORY OF MATHEMATICAL NOTATIONS
that they were warehouse marks, expressing excess or deficiency in
weights of barrels of goods.1 M. W. Drobisch,2 who was the first to
point out the occurrence of the signs + and — in Widman, says that
Widman uses them in passing, as if they were sufficiently known,
merely remarking, "Was — ist das ist minus vnd das + das ist mer."
C. I. Gerhardt,3 like De Morgan, says that the + and — were de-
rived from mercantile practice.
But Widman assigned the two symbols other significations as
well. In problems which he solved by false position the error has the
+ or — sign prefixed.4 The — was used also to separate the terms of a
proportion. In "11630-198 4610-78" it separates the first and
second and the third and fourth terms. The "78" is the computed
term, the fractional value of the fourth term being omitted in the
earlier editions of Widman's arithmetic. The sign + occurs in the
heading "Regula augmenti + decrement!" where it stands for the
Latin et ("and"), and is not used there as a mathematical symbol. In
another place Widman gives the example, "Itm eyner hat kaufft 6
eyer— 2 ^ pro 4 ^+1 ey" ("Again, someone has bought 6 eggs —
2 ^ for 4 ^ + 1 egg"), and asks for the cost of one egg. Here the — is
simply a dash separating the words for the goods from the price.
From this and other quotations Glaisher concludes that Widman
used + and — "in all the ways in which they are used in algebra."
But we have seen that Widman did not restrict the signs to that usage;
the + was used for "and" when it did not mean addition; the — was
used to indicate separation. In other words, Widman does not re-
strict the use of + and — to the technical meanings that they have in
algebra.
203. In an anonymous manuscript,5 probably written about the
time when Widman's arithmetic appeared, use is made of symbolism
in the presentation of algebraic rules, in part as follows:
"Conditiones circa + vel — in additione
+ et +\ - . +\ addalur non sumendo respectwn quis numerus sit
/facit >
— et — / — / superior.
1 De Morgan, op. cit., Vol. XI, p. 206.
2 De Joannis Widmanni .... compendia (Leipzig, 1840), p. 20 (quoted by
Glaisher, op. cit., p. 9).
*Geschichte der Mathematik in Deutschland (1877), p. 36: ". . . . dass diese
Zeichen im kaufmilnnischen Verkehr ublich waren."
4 Glaisher, op. cit., p. 15.
5 Regidae Cosae vel Algebrae, a Latin manuscript, written perhaps about 1450,
but "surely before 1510," in the Vienna Library.
ADDITION AND SUBTRACTION 235
0. f .. I + et — \ simpliciter subtrahatur minor numerus a
Siiuent { , , > • • , -i MX A in
I — et +/ majon et residue sua ascribatur nota, *
and similarly for subtraction. This manuscript of thirty-three leaves
is supposed to have been used by Henricus Grammateus (Heinrich
Schreiber) in the preparation of his Rechenbuch of 1518 and by Chris-
toff Rudolff in his Coss of 1525.
Grammateus2 in 1518 restricts his use of + and — to technical
algebra: "Vnd man braucht solche zaichen als + 1st vnnd, — myn-
nder" ("And one uses such signs as + [which] is 'and/ ~ 'less' ").
See Figure 56 for the reproduction of this passage from the edition of
1535. The two signs came to be used freely in all German algebras,
particularly those of Grammateus, Rudolff (1525), Stifel (1544), and
in Riese's manuscript algebra (1524). In a text by Eysenhut3 the +
is used once in the addition of fractions; both + and — are employed
many times in the regula falsi explained at the end of the book.
Arithmetics, more particularly commercial arithmetics, which did
not present the algebraic method of solving problems, did not usually
make use of the + and — symbols. L. L. Jackson says: "Although
the symbols + and — were in existence in the fifteenth century, and
appeared for the first time in print in Widrnan (1489), as shown in the
illustration (p. 53), they do not appear in the arithmetics as signs of
operation until the latter part of the sixteenth century. In fact, they
did not pass from algebra to general use in arithmetic until the nine-
teenth century. "4
204. Spread of the + and — symbols. — In Italy the symbols p
and m served as convenient abbreviations for "plus" and "minus"
at the end of the fifteenth century and during the sixteenth. In 1608
the German Clavius, residing in Rome, used the + and — in his
algebra brought out in Rome (see Fig. 66). Camillo Gloriosi adopted
them in his Ad theorema geometricum of 1613 and in his Exercitationes
mathematicae, decas I (Naples, 1627) (§ 196). The + and — signs were
used by B. Cavalieri (see Fig. 86) as if they were well known. The +
1 C. I. Gerhardt, "Zur Geschichte der Algebra in Deutschland," Monats-
berichte der k. pr. Akademie d. Wissenschaften z. Berlin (1870), p. 147.
2 Henricus Grammateus, Ayn New Kunsllich Buech (Niirnberg: Widrnung,
1518; publication probably in 1521). See Glaisher, op. cit., p. 34.
3 Ein kunsllich rechenbuch auff Zyffern / Lini vnd Wdlschen Practica (Augs-
burg, 1538). This reference is taken from Tropfke, op. cit., Vol. I (2d ed., 1921),
p. 58.
4 The Educational Significance of Sixteenth Century Arithmetic (New York,
1906), p. 54.
236 A HISTORY OF MATHEMATICAL NOTATIONS
and — were used in England in 1557 by Robert Recorde (Fig. 71) and
in Holland in 1637 by Gillis van der Hoecke (Fig. 60). In France and
Spain the German + and — , and the Italian p and m, came in sharp
competition. The German Scheubel in 1551 brought out at Paris an
algebra containing the + and — (§158); nevertheless, the p and m
(or the capital letters P, M) were retained by Peletier (Figs. 80, 81),
Buteo (Fig. 82), and Gosselin (Fig. 83). But the adoption of the Ger-
man signs by Ramus and Vieta (Figs. 84, 85) brought final victory for
them in France. The Portuguese P. Nunez (§ 166) used in his algebra
(published in the Spanish language) the Italian p and m. Before this,
Marco Aurel,1 a German residing in Spain, brought out an algebra at
Valencia in 1552 which contained the + and — and the symbols for
powers and roots found in Christoff Rudolff (§ 165). But ten years
later the Spanish writer P6rez de Moya returned to the Italian sym-
bolism2 with its p and w, and the use of n., co., ce, cu, for powers and
r, rr, rrr for roots. Moya explains: "These characters I am moved to
adopt, because others are not to be had in the printing office."3 Of
English authors4 we have found only one using the Italian signs for
"plus" and "minus," namely, the physician and mystic, Robert Fludd,
whose numerous writings were nearly all published on the Continent.
Fludd uses -P and M for "plus" and "minus."
The + and — , and the p and m, were introduced in the latter part
of the fifteenth century, about the same time. They competed with
each other for more than a century, and p and m finally lost out in the
early part of the seventeenth century.
205. Shapes of the plus sign. — The plus sign, as found in print, has
had three principal varieties of form: (1) the Greek cross +, as it is
found in Widman (1489); (2) the Latin cross, T more frequently
placed horizontally, — |- or H — ; (3) the form *J«, or occasionally some
form still more fanciful, like the eight-pointed Maltese cross *%*, or a
cross having four rounded vases with tendrils drooping from their
edges.
The Greek cross, with the horizontal stroke sometimes a little
1 Libro primero de Arithmetica Algebratica .... por Marco Aurel, natural
Alcman (Valencia, 1552).
2 J. Rey Pastor, Los mathemdticos cspaiioles del siglo XVI (Oviedo, 1913), p. 38.
8 "Estos characteres me ha parecido poner, porque no auia otros eri la im-
prenta" (Ad theorema geometricvm, d nobilissimo viro propositum, Joannis Camilli
Gloriosi responsum [Venctiis, 1613], p. 26).
4 See C. Henry, Revue archeologigue, N.S., Vol. XXXVII, p. 329, who quotes
from Fludd, Utriusque cosmi .... Historia (Oppenheim, 1617).
ADDITION AND SUBTRACTION 237
longer than the vertical one, was introduced by Widman and has
been the prevailing form of plus sign ever since. It was the form com-
monly used by Grammateus, Rudolff, Stifel, Recorde, Digges, Clavius,
Dee, Harriot, Oughtred, Rahn, Descartes, and most writers since their
time.
206. The Latin cross, placed in a horizontal position, thus — )-,
was used by Vieta1 in 1591. The Latin cross was used by Romanus,2
Hunt,3 Hume,4 Hdrigone,6 Mengoli,6 Huygens,7 Ferrnat,8 by writers in
the Journal des Sgavans,9 Dechales,10 Rolle,11 Lamy,12 L'Hospital,13
Swedenborg,14 Pardies,16 Kresa,16 Belidor,17 De Moivre,18 and Michel-
sen.19 During the eighteenth century this form became less common
and finally very rare.
Sometimes the Latin cross receives special ornaments in the form
of a heavy dot at the end of each of the three shorter arms, or in the
form of two or three prongs at each short arm, as in H. Vitalis.20 A
very ostentatious twelve-pointed cross, in which each of the four equal
1 Vieta, In artem analyticam isagoge (Turonis, 1591).
2 Adriani Romani Canon triangvlorvm sphaericorum .... (Mocvntiae, 1609).
3 Nicolas Hunt, The Hand-Maid to Arithmetick (London, 1633), p. 130.
4 James Hume, Traite de Valgebre (Paris, 1635), p. 4.
6 P. Herigone, "Explicatis notarvm," Cvrsvs mathematicvs, Vol. I (Paris, 1634).
6 Petro Merigoli, Geometriae speciosae elementa (Bologna, 1659), p. 33.
7 Chrisliani Hvgenii Holorogivm oscillatorivm (Paris, 1673), p. 88.
8 P. de Ferniat, Diophanli Alexandrini Arithmeticorum libri sex (Toulouse,
1670), p. 30; see also Fermat, Varia opera (1679), p. 5.
9 Op. cit. (Amsterdam, 1680), p. 160; ibid. (1693), p. 3, and other places.
10 K. P. Claudii Francisci Milliet Dechales, Mundus mathcmaticus, Vol. I
(Leyden, 1690), p. 577.
11 M. Rolle, Methode pour resoudre les egalitez de tons Us degreez (Paris, 1691),
p. 15.
12 Bernard Lamy, Siemens des mathematiques (3d ed.; Amsterdam, 1692), p. 61.
13 L'JTospital, Ada eruditorum (1694), p. 194; ibid. (1695), p. 59; see also other
places, for instance, ibid. (1711), SuppL, p. 40.
14 Kmanuel Swedenborg, Daedalus Hyperborens (Upsala, 1716), p. 5; reprinted
in Kungliga Vetenskaps Societetens i Upsala Tvdhundr adrsminne (1910).
15 (Euvres du R. P. Pardies (Lyon, 1695), p. 103.
16 J. Kresa, Analysis speciosa trigonometriae sphericae (Prague, 1720), p. 57.
17 B. F. de Belidor, Nouveau cours de mathematique (Paris, 1725), p. 10.
18 A. de Moivre, Miscellanea analytica (London, 1730), p. 100.
19 J. A. C. Michelsen, Theorie der Gleichungen (Berlin, 1791).
20 "Algebra," Lexicon mathematicum authore Hieronymo Vitali (Rome, 1690).
238 A HISTORY OF MATHEMATICAL NOTATIONS
arms has three prongs, is given by Carolo Renaldini.1 In seventeenth-
and eighteenth-century books it is not an uncommon occurrence to
have two or three forms of plus signs in one and the same publication,
or to find the Latin cross in an upright or horizontal position, accord-
ing to the crowded condition of a particular line in which the symbol
occurs.
207. The cross of the form *%• was used in 1563 and earlier by the
Spaniard De Hortega,2 also by Klcbotius,3 Romanus,4 and Des-
cartes.5 It occurs not infrequently in the Ada eruditorum6 of Leipzig,
and sometimes in the Miscellanea Berolinensia.7 It was sometimes
used by Halley,8 Weigel,9 Swedenborg,10 and Wolff.11 Evidently this
symbol had a wide geographical distribution, but it never threatened
to assume supremacy over the less fanciful Greek cross.
A somewhat simpler form, + , consists of a Greek cross with four
uniformly heavy black arms, each terminating in a thin line drawn
across it. It is found, for example, in a work of Hindenburg,12 and
renders the plus signs on a page unduly conspicuous.
Occasionally plus signs are found which make a "loud" display
on the printed page. Among these is the eight-pointed Maltese cross,
1 Car oli Renaldini Ars analytica mathematicvm (Florence, 1665), p. 80, and
throughout the volume, while in the earlier edition (Anconnae, 1644) he uses both
the heavy eross and dagger form.
2 Fray Jua dc Hortega, Tractado subtilissimo d' arismetica 7 geometria (Gra-
nada, 1563), leaf 51. Also (Seville, 1552), leaf 42.
3 Guillaume Klebitius, Insvlae Melitensis, quam alias Maltam vocant, Historia,
Quacstionib. aliquot Mathcmalicis rcddila incundior (Diest [Belgium], 1565). I
arn indebted to Professor H. Bosnians for information relating to this book.
4 Adr. Romanus, "Problema," Ideae malhematicae pars prima (Antwerp, 1593).
5 Ilene Descartes, La geometric (1637), p. 325. This form of the plus sign is in-
frequent in this publication; the ordinary form (-f ) prevails.
6 See, for instance, op. cit. (1682), p. 87; ibid. (1683), p. 204; ibid. (1691),
p. 179; ibid. (1694), p. 195; ibid. (1697), p. 131; ibid. (1698), p. 307; ibid. (1713),
p. 344.
7 Op. cit., p. 156. However, the Latin cross is used more frequently than the
form now under consideration. But in Vol. 11 (1723), the latter form is prevalent.
8 K Halley, Philosophical Transactions, Vol. XVII (London, 1692-94), p. 963;
ibid. (1700-1701), Vol. XXII, p. 625.
9 Erhardi Weigelii Philosophia Mathematica (Jena, 1693), p. 135.
10 E. Swedenborg, op. cit., p. 32. The Latin cross is more prevalent in this
book.
11 Christian Wolff, Mathcmatisches Lexicon (Leipzig, 1716), p. 14.
12 Carl Friedrich Hindenburg, Injinitinomii dignitaium .... leges ac Formulae
(Gottingen, 1779).
ADDITION AND SUBTRACTION 239
of varying shape, found, for example, in James Gregory,1 Corachan,2
Wolff,3 and Hindenburg.4
Sometimes the ordinary Greek cross has the horizontal stroke
very much heavier or wider than the vertical, as is seen, for instance,
in Fortunatus.5 A form for plus — / — occurs in Johan Albert.6
208. Varieties of minus signs. — One of the curiosities in the his-
tory of mathematical notations is the fact that notwithstanding the
extreme simplicity and convenience of the symbol — to indicate sub-
traction, a more complicated symbol of subtraction -s- should have
been proposed and been able to maintain itself with a considerable
group of writers, during a period of four hundred years. As already
shown, the first appearance in print of the symbols + and — for
"plus" and "minus" is found in Widman's arithmetic. The sign — is
one of the very simplest conceivable; therefore it is surprising that a
modification of it should ever have been suggested.
Probably these printed signs have ancestors in handwritten docu-
ments, but the line of descent is usually difficult to trace with cer-
tainty (§ 201). The following quotation suggests another clue: "In
the west-gothic writing before the ninth century one finds, as also
Paoli remarks, that a short line has a dot placed above it — , to indi-
cate my in order to distinguish this mark from the simple line which
signifies a contraction or the letter N. But from the ninth century
down, this same wcst-gothic script always contains the dot over the
line even when it is intended as a general mark."7
In print the writer has found the sign — for "minus" only once.
It occurs in the 1535 edition of the Rechenbuchlin of Grammateus
(Fig. 56). He says: "Vnd man brauchet solche zeichen als + ist
mehr / vnd — / minder."8 Strange to say, this minus sign does not
occur in the first edition (1518) of that book. The corresponding pas-
sage of the earlier edition reads: "Vnd man braucht solche zaichen
1 Geometriae pars vniversalis (Padua, 1668), p. 20, 71, 105, 108.
2 Juan Bautista Corachan, Arithmetica demonstrada (Barcelona, 1719), p. 326.
3 Christian Wolff, Elementa matheseos universae, Tomus I (Halle, 1713), p. 252.
4 Op. cit.
6 P. F. Fortunatus, Elementa matheseos (Brixia, 1750), p. 7.
8 Johan Albert, New Rechenbuchlein auff der federn (Wittembcrg, 1541);
taken from Glaisher, op. cit., p. 40, 61.
7 Adriano Cappelli, Lexicon abbreviaturam (Leipzig, 1901), p. xx.
8 Henricus Grammateus, Eyn new Kunstlich behend and gewiss Rechenbuchlin
(1535; 1st ed., 1518). For a facsimile page of the 1535 edition, see D. E. Smith,
Kara arithmetica (1908), p. 125.
240 A HISTORY OF MATHEMATICAL NOTATIONS
als + ist vnnd / — mynnder." Nor does Grammateus use ~ in other
parts of the 1535 edition; in his mathematical operations the minus
sign is always — .
The use of the dash and two dots, thus -T-, for "minus," has been
found by Glaisher to have been used in 1525, in an arithmetic of
Adam Riese,1 who explains: "Sagenn sie der warheit zuuil so be-
zeychenn sie mit dem zeychen + plus wu aber zu wenigk so beschreib
sie mit dem zeychen -5- minus genant."2
No reason is given for the change from — to -5-. Nor did Riese
use -5- to the exclusion of — . He uses -f- in his algebra, Die Coss, of
1524, which he did not publish, but which was printed3 in 1892, and
also in his arithmetic, published in Leipzig in 1550. Apparently, he
used — more frequently than -f- .
Probably the reason for using -*- to designate — lay in the fact
that — was assigned more than one signification. In Widman's
arithmetic — was used for subtraction or "minus," also for separating
terms in proportion,4 and for connecting each amount of an article
(wool, for instance) with the cost per pound (§ 202). The symbol —
was also used as a rhetorical symbol or dash in the same manner as it
is used at the present time. No doubt, the underlying motive in
introducing -f- in place of — was the avoidance of confusion. This
explanation receives support from the German astronomer Regio-
montanus,5 who, in his correspondence with the court astronomer at
Ferrara, Giovanni Bianchini, used — as a sign of equality; and used
for subtraction a different symbol, namely, ip (possibly a florescent
form of m). With him 1 ip re meant 1 — x.
Eleven years later, in 1546, Gall Splenlin, of Ulm, had published
at Augsburg his Arithmetica kunstlicher Rechnung, in which he uses -5-,
saying: "Bedeut das zaichen -f zuuil, und das •*- zii wenig."6 Riese
and Splenlin are the only arithmetical authors preceding the middle
of the sixteenth century whom Glaisher mentions as using -~ for sub-
traction or "minus."7 Caspar Thierfeldern,8 in his Arithmetica
1 Rechenung auff der linihen vndfedern in zal, masz, vnd gewicht (Erfurt, 1525;
1st ed., 1522).
2 This quotation is taken from Glaisher, op. cit., p. 36.
3 See Bruno Berlet, Adam Riese (Leipzig, Frankfurt am Main, 1892).
4 Glaisher, op. cit.t p. 15.
6 M. Curtze, Abhandlungen zur Geschichte der mathematischen Wissenschaften,
Vol. XII (1902), p. 234; Karpinski, Robert of Chester, etc., p. 37.
6 See Glaisher, op. cit.t p. 43.
7 Ibid., Vol. LI, p. 1-148. 8 See Jackson, op. cit., p. 55, 220.
ADDITION AND SUBTRACTION 241
(Nuremberg, 1587), writes the equation (p. 110), "18 fl.-v-85 gr.
gleich25fl.-r-232gr."
With the beginning of the seventeenth century -f- for "minus"
appears more frequently, but, as far as we have been able to ascertain
only in German, Swiss, and Dutch books. A Dutch teacher, Jacob
Vander Schuere, in his Arithmetica (Haarlem, 1600), defines + and
— , but lapses into using -4- in the solution of problems. A Swiss
writer, Wilhelm Schey,1 in 1600 and in 1602 uses both -r- and TT for
"minus." He writes 9+9, 54-12, 6-f-28, where the first number sig-
nifies the weight in centner and the second indicates the excess or
deficiency of the respective "pounds." In another place Schey writes
"9 fl. -rr 1 ort," which means "9 florins less 1 ort or quart." In 1601
Nicolaus Reymers,2 an astronomer and mathematician, uses regularly
•f- for "minus" of subtraction; he writes
"XXVIII XII X VI III I 0
Igr. 65532+18 -v-30 -18 +12-5-8"
for z28=65,532z12+18z10-30z6-18z3+12z-~8 .
Peter Roth, of Niirnberg, uses 44- in writing3 3x2 — 26z. Johannes
Faulhaber4 at Ulm in Wiirttemberg used -5- frequently. With him the
horizontal stroke was long and thin, the dots being very near to it.
The year following, the symbol occurs in an arithmetic of Ludolf
van Ceulen,6 who says in one place: "Subtraheert |/7 van, \/13, rest
1/13, weynigher j/7, daerom stelt |/13 voren en j/7 achter, met een
sulck teecken -*- tusschen beyde, vvelck teecmin beduyt, comt alsoo de
begeerde rest j/13-r-j/7 — ." However, in some parts of the book —
is used for subtraction. Albert Girard6 mentions -5- as the symbol for
"minus," but uses — . Otto Wesellow7 brought out a book in which
1 Arithmetica oder die Kunst zu rechnen (Basel, 1600-1602). We quote from
D. E. Smith, op. cit., p. 427, and from Matthiius Sterner, Geschichte der Rechen-
kunst (Munchen and Leipzig, 1891), p. 280, 291.
2 Nicolai Raimari Ursi Dithmarsi .... arithmetica analytica, vulgo Cosa, oder
Algebra (zu Frankfurt an der Oder, 1601). We take this quotation from Gerhardt,
Geschichte der Mathematik in Deutschland (1877), p. 85.
3 Arithmetica philosophica (1608). We quote from Treutlein, "Die deutsche
Coss," Abhandlungen zur Geschichte der Mathematik, Vol. II (Leipzig, 1879),
p. 28, 37, 103.
4 Numerus figuratus sive arithmetica analytica (Ulm, 1614), p. 11, 16.
6 De arithmetische en geometrische Fondamenten (1615), p. 52, 55, 56.
6 Invention nouvelle en Valgebre (Amsterdam, 1629), no paging. A facsimile
edition appeared at Leiden in 1884,
7 Flores arithmetici (driidde vnde veerde deel; Bremen, 1617), p. 523.
242 A HISTORY OF MATHEMATICAL NOTATIONS
+ and -4- stand for "plus" and "minus," respectively. These signs
are used by Follinus,1 by Stampioen (§ 508), by Daniel van Hovcke2
who speaks of + as 'signifying "mer en -f- min.," and by Johann
Ardiiser3 in a geometry. It is interesting to observe that only thirteen
years after the publication of Ardiiser's book, another Swiss, J. H.
Rahn, finding, perhaps, that there existed two signs for subtraction,
but none for division, proceeded to use -f- to designate division. This
practice did not meet with adoption in Switzerland, but was seized
upon with great avidity as the symbol for division in a far-off country,
England. In 1670 -r was used for subtraction once by Huygens4 in
the Philosophical Transactions. Johann Hemelings5 wrote -H- for
"minus" and indicated, in an example, 14^ legions less 1250 men by
"14 1/2 Legion -f- 1250 Mann." The symbol is used by Tobias
Beutel,6 who writes "81^1fi6561-f-162. R. + l. zenss" to represent
our 81 — 1/6561 — 162#+£2. Kegel7 explains how one can easily
multiply by 41, by first multiplying by 6, then by 7, and finally sub-
tracting the multiplicand; he writes "7-7-1." In a set of seventeenth-
century examination questions used at Ntirnberg, reference is made
to cossic operations involving quantities, "durch die Signa + und -f-
connectirt."8
The vitality of this redundant symbol of subtraction is shown by
its continued existence during the eighteenth century. It was em-
ployed by Paricius,9 of Regensburg. Schlesser10 takes ^ to represent
1 Hermannus Follinus, Algebra sive liber de rebus occultis (Coloniae, 1622),
p. 113, 185.
2Cyffer-Boeck .... (den tweeden Druck: Rotterdam, 1628), p. 129-33.
8 Geometriae theoricae et practicae. Oder von dem Feldmdssen (Zurich, 1646),
fol. 75.
4 In a reply to Slusius, Philosophical Transactions, Vol. V (London, 1670), p.
6144.
5 Arithmetisch-Poetisch-u. Historisch-Erquick Stund (Hannover, 1660) ; Selbst-
lehrendes Rechen-Buch .... durch Johannem Hemelingium (Frankfurt, 1678).
Quoted from Hugo Grosse, Historische Rechenbucher des 16. and 17. Jahrhunderts
(Leipzig, 1901), p. 99, 112.
6 Geometrische Gallerie (Leipzig, 1690), p. 46.
7 Johann Michael Kegel, New vermehrte arithmetica vulgaris et practica italica
(Frankfurt am Main, 1696). We quote from Sterner, op. cit., p. 288.
8 Fr. Unger, Die Methodik der praktischen Arithmetik in historischer Ent-
wickelung (Leipzig, 1888), p. 30.
9 Georg Heinrich Paricius, Praxis arithmetices (1706). We quote from Sterner,
op. cit., p. 349.
10 Christian Schlesser, Arithmetisches Haupt-Schliissel .... Die Coss — oder
Algebra (Dresden and Leipzig, 1720).
ADDITION AND SUBTRACTION 243
"minus oder weniger." It was employed in the Philosophical Transac-
tions by the Dutch astronomer N. Cruquius;1 -*- is found in Hubsch2
and Crusius.3 It was used very frequently as the symbol for subtrao
tion and "minus" in the Maandelykse Mathematische Liefhebbery,
Purmerende (1754-69). It is found in a Dutch arithmetic by Bartjens4
which passed through many editions. The vitality of the symbol is dis-
played still further by its regular appearance in a book by van Steyn,6
who, however, uses — in 1778.6 Halcke states, "-f- of — het teken
van substractio minims of min.,"7 but uses — nearly everywhere. Praal-
der, of Utrecht, uses ordinarily the minus sign — , but in one place8 he
introduces, for the sake of clearness, as he says, the use of -f- to mark
the subtraction of complicated expressions. Thus, he writes
" = -v-9^+2j/26." The -f- occurs in a Leipzig magazine,9 in a Dresden
work by Illing,10 in a Berlin text by Schmeisser,11 who uses it also in
expressing arithmetical ratio, as in "2-f-6-$- 10." In a part of KliigePs12
mathematical dictionary, published in 1831, it is stated that -s- is
used as a symbol for division, "but in German arithmetics is employed
also to designate subtraction." A later use of it for "minus," that we
have noticed, is in a Norwegian arithmetic.13 In fact, in Scandinavian
1 Op. tit., Vol. XXXIII (London, 1726), p. 5, 7.
2 J. G. G. Hiibsch, Arithmelica portensis (Leipzig, 1748).
8 David Arnold Crusius, Anweisung zur Rechen-Kunst (Halle, 1746), p. 54.
4 De vernieuwde Cyfferinge van Mr. Willvm Bartjens, .... vermeerderl — ende
verbetert, door Mr. Jan van Dam en van alle voorgaande Fauten gezuyvert door
Klaas Bosch (Amsterdam, 1771), p. 174-77.
6 Gerard van Steyn, Liefhebbery der Reekenkonst (eerste deel; Amsterdam,'
1768), p. 3, 11, etc.
6 Ibid. (2° Deels, 2« Stuk, 1778), p. 16.
7 Mathematisch Zinnen-Confect .... door Paul Halcken .... Uyt het Hoog-
duytsch vertaald . ... dor Jacob Oostwoud (Tweede Druk, Te Purmerende, 1768),
p. 5.
8 Mathematische Voorstellen .... door .... Ludolf van Keulen .... door
Laurens Praalder (Amsterdam, 1777), p. 137.
9 J. A. Kritter, Leipziger Magazin fur reine and angewandte Mathematik
(hcrausgegeberi von J. Bernoulli und C. F. Hindenburg, 1788), p. 147-61.
10 Carl Christian Illing, Arithmetisches Handbuch fur Lehrer in den Schulen
(Dresden, 1793), p. 11, 132.
11 Friedrich Schmeisser, Lehrbuch der reinen Mathesis (1. Theil, Berlin, 1817),
p. 45, 201.
18 G. S. Ktugel, "Zeichen," Mathematisches Wdrterbuch. This article was writ-
ten by J. A. Grunert.
18 G. C. Krogh, Regnebogfor Begyndere (Bergen, 1869), p. 15.
244 A HISTORY OF MATHEMATICAL NOTATIONS
countries the sign -f- for "minus" is found occasionally in the twentieth
century. For instance, in a Danish scientific publication of the year
1915, a chemist expresses a range of temperature in the words
"fra+18° C. til ^ 18° C."1 In 1921 Ernst W. Selmer2 wrote "0,72 4-
0,65 = 0,07." The difference in the dates that have been given, and the
distances between the places of publication, make it certain that this
symbol -T- for "minus" had a much wider adoption in Germany,
Switzerland, Holland, and Scandinavia than the number of our cita-
tions would indicate. But its use seems to have been confined to
Teutonic peoples.
Several writers on mathematical history have incidentally called
attention to one or two authors who used the symbol -r- for "minus,"
but none of the historians revealed even a suspicion that this symbol
had an almost continuous history extending over four centuries.
209. Sometimes the minus sign — appears broken up into two or
three successive dashes or dots. In a book of 1610 and again of 1615,
by Ludolph van Ceulen,3 the minus sign occasionally takes the form
— . Richard Balam4 uses three dots and says "3 • • • 7, 3 from 7";
he writes an arithmetical proportion in this manner: "2 • • -4 =
3 • • -5." Two or three dots are used in Ren6 Descartes' Geometric,
in the writings of Marin Mersenne,5 and in many other seventeenth-
century books, also in the Journal des S$avans for the year 1686,
printed in Amsterdam, where one finds (p. 482) "1 R 11"
for 1 — l/— il, and in volumes of that Journal printed in the early
part of the eighteenth century. Herigone used ~ for "minus"
(§ 189), the — being pre-empted for recta linea.
From these observations it is evident that in the sixteenth and
seventeenth centuries the forms of type for "minus" were not yet
standardized. For this reason, several varieties were sometimes used
on the same page.
This study emphasizes the difficulty experienced even in ordinary
1 Johannes Boye Pctcrsen, Kgl. Danske Vidensk. Selskabs Skrifter, Nat. og.
Math, Afd., 7. Raekke, Vol. XII (Kopenhagen, 1915), p. 330; sec also p. 221, 223,
226, 230, 238.
2 Skrifter utgit av Videnskapsselskapet i Kristiania (1921)," Historisk-filosofisk
Klasse" (2. Bind; Kristiania, 1922), article by Ernst W. Selmer, p. 11; see also
p. 28, 29, 39, 47.
3 Circvlo et adscriptis liber Omnia e vernaculo Latina fecit et annotationibus
illustravit Willebrordus Snellius (Leyden, 1610), p. 128.
4 Algebra (London, 1653), p. 5.
*Cogitata Physico-Mathematica (Paris, 1644), Praefatio generalis, "De
Rationibus atque Proportionibus," p. xii, xiii.
ADDITION AND SUBTRACTION 245
arithmetic and algebra in reaching a common world-language. Cen-
turies slip past before any marked step toward uniformity is made.
It appears, indeed, as if blind chance were an uncertain guide to lead
us away from the Babel of languages. The only hope for rapid ap-
proach of uniformity in mathematical symbolism lies in international
co-operation through representative committees.
210. Symbols for "plus or minus" — The ± to designate "plus or
minus" was used by Albert Girard in his Tables1 of 1626, but with the
+
interpolation of ou, thus "ou" The ± was employed by Oughtred in
his Clavis mathematicae (1631), by Wallis,2 by Jones3 in his Synopsis,
and by others. There was considerable experimentation on suitable
notations for cases of simultaneous double signs. For example, in
the third book of his Geometric, Descartes uses a dot where we would
+PP
write ±. Thus he writes the equation "+?/6«2p?/4 4ryy—qqy>Q"
and then comments on this: "Et pour les signes -f- ou — que iay
omis, s'il y a eu+p en la precedente Equation, il faut mettre en celle —
cy + 2p, ou s'il ya eu — p, il faut mettre — 2p; & au contraire s'il
ya eu + r, il faut mettre — 4r, ..." The symbolism which in the Mis-
cellanea Berolinensia of 1710 is attributed to Leibniz is given in § 198.
A different notation is found in Isaac Newton's Universal Arith-
metick: "I denoted the Signs of b and c as being indeterminate by
the Note J-, which I use indifferently for + or — , and its opposite
T for the contrary."4 These signs appear to be the + with half of
the vertical stroke excised. William Jones, when discussing quadratic
equations, says: "Therefore if V be put for the Sign of any Term,
and A for the contrary, all Forms of Quadratics with their Solutions,
will be reduced to this one. If xxVaxVb = Q then A^a±aa Ab|*."5
Later in the book (p. 189) Jones lets two horizontal dots represent
any sign: "Suppose any Equation whatever, as xn . . ax7*"-1 . . 6xn~2
. . cxn~* . . dxn~4, etc. . . A=0."
A symbol 8 standing for ± was used in 1649 and again as late as
1695, by van Schooten6 in his editions of Descartes' geometry, also
1 See Bibliotheca mathematica (3d ser., 1900), Vol. I, p. 66.
2 J. Wallis, Operum mathematicorum pars prima (Oxford, 1657), p. 250.
3 William Jones, Synopsis Palmariorum matheseos (London, 1706), p. 14.
4 Op. cit. (trans. Mr. Ralphson .... rev. by Mr. Cunn; London, 1728), p. 172;
also ibid. (rev. by Mr. Cunn .... expl. by Theaker Wilder; London, 1769), p. 321.
6 Op. cit., p. 148.
6 Renati Descartes Geometria (Leyden, 1649), Appendix, p. 330; ibid. (Frank-
furt am Main, 1695), p. 295, 444, 445.
246 A HISTORY OF MATHEMATICAL NOTATIONS
by De Witt.1 Wallis2 wrote & for + or — , and R for the contrary.
The sign & was used in a restricted way, by James Bernoulli;3 h6
says, "% significat + in pr. e — in post, hypoth.," i.e., the symbol
stood for + according to the first hypothesis, and for — , according to
the second hypothesis. He used this same symbol in his Ars con-
jectandi (1713), page 264. Van Schooten wrote also # for + . It
should be added that tf appears also in the older printed Greek books
as a ligature or combination of two Greek letters, the omicron o and
the upsilon v. The tf appears also as an astronomical symbol for the
constellation Taurus.
Da Cunha4 introduced ±f and ±', or ±' and +', to mean that
the upper signs shall be taken simultaneously in both or the lower
signs shall be taken simultaneously in both. Oliver, Wait, and Jones5
denoted positive or negative N by *N.
211. The symbol [a] was introduced by Kronecker6 to represent
0 or + 1 or — 1, according as a was 0 or + 1 or — 1. The symbol "sgn"
has been used by some recent writers, as, for instance, Peano,7 Netto,8
and Le Vavasseur, in a manner like this: "sgn A = +1" when A >0,
"sgn A = —l" when A<0. That is, "sgn A" means the "sign of
A." Similarly, Kowalewski9 denotes by "sgn $" +1 when $ is an
even, and — 1 when ^ is an odd, permutation.
The symbol I/a2 is sometimes taken in the sense10 ± a, but in equa-
tions involving i/ , the principal root +a is understood.
212. Certain other specialized uses of + and — . — The use of each
of the signs + and — in a double sense— first, to signify addition and
subtraction; second, to indicate that a number is positive and nega-
tive— has met with opposition from writers who disregarded the ad-
vantages resulting from this double use, as seen in a— (— 6)=a+6,
1 Johannis de Witt, Elementa Cvrvarvm Linearvm. Edita Opera Francisci a
Schooten (Amsterdam, 1683), p. 305.
2 John Wallis, Treatise of Algebra (London, 1685), p. 210, 278.
3 Ada eruditorum (1701), p. 214.
4 J. A. da Cunha, Principles mathematicos (Lisbon, 1790), p. 126.
B Treatise on Algebra (2d cd.; Ithaca, 1887), p. 45.
6 L. Kronecker, Werke, Vol. II (1897), p. 39.
7 G. Peano, Formulario mathematico, Vol. V (Turin, 1908), p. 94.
8 E. Netto and R. le Vavasseur, Encyclopedic des scien. math,, Tome I, Vol. II
(1907), p. 184; see also A. Voss and J. Molk, ibid., Tome II, Vol. I (1912), p. 257,
n. 77.
9 Gerhard Kowalew8kitEinfuhrungindieDeterminantentheorie (Leipzig, 1909),
p. 18.
10 See, for instance, Encyclopedic des scien. math., Tome II, Vol. I, p. 257, n. 77.
ADDITION AND SUBTRACTION 247
and who aimed at extreme logical simplicity in expounding the ele-
ments of algebra to young pupils. As a remedy, German writers
proposed a number of new symbols which are set forth by Schmeisser
as follows:
"The use of the signs + and — , not only for opposite magni-
tudes .... but also for Addition and Subtraction, frequently pre-
vents clearness in these matters, and has even given rise to errors.
For that reason other signs have been proposed for the positive and
negative. Wilkins (Die Lehre von d. entgegengesetzL Grossen etc.,
Brschw., 1800) puts down the positive without signs (+a = a) but
places over the negative a dash, as in — a=d. v. Winterfeld (An*
fangsgr. d. Rechenk., 2te Aufl. 1809) proposes for positive the sign h
or f, for negative H or ~|. As more scientific he considers the in-
version of the letters and numerals, but unfortunately some of them
as iy r, o, x, etc., and 0, 1, 8, etc., cannot be inverted, while others, by
this process, give rise to other letters as &, d, p, q, etc. Better are the
more recent proposals of Winterfeld, to use for processes of computa-
tion the signs of the waxing and waning moon, namely for Addition
), for Subtraction (, for Multiplication }, for Division <(, but as he
himself acknowledges, even these are not perfectly suitable
Since in our day one does not yet, for love of correctness, abandon the
things that are customary though faulty,, it is for the present probably
better to stress the significance of the concepts of the positive and
additive, and of the negative and subtractive, in instruction, by the
retention of the usual signs, or, what is the same thing, to let the
qualitative and quantitative significance of + and — be brought out
sharply. This procedure has the advantage moreover of more fully
exercising the understanding. "l
Wolfgang Bolyai2 in 1832 draws a distinction between + and — ,
and + and H-» ; the latter meaning the (intrinsic) "positive" and
"negative." If A signifies *-*B, then —A signifies +#.
213. In more recent time other notations for positive and nega-
tive numbers have been adopted by certain writers. Thus, Spitz3
uses <-a and ->a for positive a and negative a, respectively. M^ray4
prefers "a , "a; Pad6,5 ap, an; Oliver, Wait, and Jones6 employ an ele-
1 Friedrieh Schmeisser, op. cit.} p. 42, 43.
2 Tentamen (2d ed., T. L; Budapestini, 1897), p. xi.
* C. Spitz, Lehrbuch der alg. Arilhmetik (Leipzig, 1874), p. 12.
4 Charles M£ray, Lemons nouv. de V analyse infin., Vol. I (Paris, 1894), p. 11.
6 H. Fade*, Premieres legons d'algtbre &&m. (Paris, 1892), p. 5.
9 Op. cit., p. 5.
248 A HISTORY OF MATHEMATICAL NOTATIONS
vated + or — (as in +10, -10) as signs of "quality"; this practice has
been followed in developing the fundamental operations in algebra by
a considerable number of writers; for instance, by Fisher and Schwatt,1
and by Slaught and Lennes.2 In elementary algebra the special sym-
bolisms which have been suggested to represent "positive number"
or "negative number" have never met with wide adoption. Stolz
and Gmeiner3 write a, a, for positive a and negative a. The designa-
tion .... ~3, ~2, -1, 0, +1, +2, +3, . . . . , occurs in Huntington's
Continuum (1917), page 20.
214. A still different application of the sign + has been made in
the theory of integral numbers, according to which Peano4 lets a+
signify the integer immediately following a, so that a+ means the inte-
ger (a+ 1). For the same purpose, Huntington5 and Stolz and Gmeiner6
place the + in the position of exponents, so that 5+ = 6.
215. Four unusual signs. — The Englishman Philip Ronayne used
in his Treatise of Algebra (London, 1727; 1st ed., 1717), page 4, two
curious signs which he acknowledged were "not common," namely,
the sign -e to denote that "some Quantity indefinitely Less than the
Term that next precedes it, is to be added," and the sign e- that such
a quantity is "to be subtracted," while the sign J> may mean "either
-e or e- when it matters not which of them it is." We have not noticed
these symbols in other texts.
How the progress of science may suggest newr symbols in mathe-
matics is illustrated by the composition of velocities as it occurs in
Einstein's addition theorem.7 Silberstein uses here # instead of +•
216. Composition of ratios. — A strange misapplication of the +
sign is sometimes found in connection with the "composition" of
NP AN
ratios. If the ratios -^jrr and -^^ are multiplied together, the product
1 G. E. Fisher and I. J. Schwatt, Text-Book of Algebra (Philadelphia, 1898),
p. 23.
2 H. E. Slaught and U. J. Lennes, High School Algebra (Boston, 1907), p. 48.
1 Otto Stolz und J. A. Gmeiner, Theoretische Arithmetik (2d ed. ; Lej'pzig, 1911),
Vol. I, p. 116.
4G. Peano, Arithmetices principia nova methodo exposita (Turin, 1889);
"Sul concetto di numero," Rivista di matem., Vol. I, p. 91; Formulaire de mathe-
matiques, Vol. II, § 2 (Turin, 1898), p. 1.
8 E. V. Huntington, Transactions of the American Mathematical Society, Vol. VI
(1905), p. 27.
6 Op. cit., Vol. I, p. 14. In the first edition Peano's notation was used.
7 C. E. Weatherburn, Advanced Vector Analysis (London, 1924), p. xvi.
ADDITION AND SUBTRACTION 249
NP AN
?T\f'rW> accorcling to an old phraseology, was "compounded" of the
L/./V C-iV
first two ratios.1 Using the term "proportion" as synonymous with
"ratio," the expression "composition of proportions" was also used.
As the word "composition" suggests addition, a curious notation,
using +, was at one time employed. For example, Isaac Barrow2 de-
NP AN
noted the "compounded ratio" 77^7 'T^F in this manner, "ArP«CAr+
C./V C./V
AN'CN." That is, the sign of addition was used in place of a sign of
multiplication, and the dot signified ratio as in Oughtred.
In another book3 Barrow again multiplies equal ratios by equal
ratios. In modern notation, the two equalities are
(PL+QO):QO = 2BC:(BC-CP) and QO:BC = BC:(BC+CP) .
Barrow writes the result of the multiplication thus:
PL+QO.QO+QO-BC=2BC-BC-CP+BC-BC+CP .
Here the + sign occurs four times, the first and fourth times as a
symbol of ordinary addition, while the second and third times it
occurs in the "addition of equal ratios" which really means the multi-
plication of equal ratios. Barrow's final relation means, in modern
notation,
BC
^ _
QO " * BC BC-CP ' BC+CP '
Wallis, in his Treatise of Algebra (London, 1685), page 84, com-
ments on this subject as follows: "But now because Euclide gives to
this the name of Composition, which word is known many times to im-
part an Addition; (as when we say the Line ABC is compounded of AB
and BC;) some of our more ancient Writers have chanced to call it
Addition of Proportions; and others, following them, have continued
that form of speech, which abides in (in divers Writers) even to this
day: And the Dissolution of this composition they call Subduction of
Proportion. (Whereas that should rather have been called Multi-
plication, and this Division.)"
A similar procedure is found as late as 1824 in J. F. Lorenz' trans-
1 See Euclid, Elements, Book VI, Definition 5. Consult also T. L. Heath, The
Thirteen Books of Euclid's "Elements," Vol. II (Cambridge, 1908), p. 132-35, 189,
190.
2 Lectiones opticae (1669), Lect. VIII, § V, and other places.
1 Lectiones geometricae (1674), Lect. XI, Appendix I, § V.
250 A HISTORY OF MATHEMATICAL NOTATIONS
lation from the Greek of Euclid's Elements (ed. C. B. Mollweide;
Halle, 1824), where on page 104 the Definition 5 of Book VI is given
thus: "Of three or more magnitudes, A} B> C, D, which are so related
to one another that the ratios of any two consecutive magnitudes
A : B, B:C, C : D, are equal to one another, then the ratio of the first
magnitude to the last is said to be composed of all these ratios so that
:C) + (C:D)" An modern notation, ~ = ™
• ~
SIGNS OF MULTIPLICATION
217. Early symbols. — In the early Babylonian tablets there is,
according to Hilprecht,1 an ideogram A-DU signifying "times" or
multiplication. The process of multiplication or division was known
to the Egyptians2 as wshtp, "to incline the head"; it can hardly be
regarded as being a mathematical symbol. Diophantus used no
symbol for multiplication (§ 102). In the Bakhshali manuscript
multiplication is usually indicated by placing the numbers side by
side (§ 109). In some manuscripts of Bhaskara and his commentators
a dot is placed between factors, but without any explanation (§ 112).
The more regular mark for product in Bhaskara is the abbreviation
bha, from bhavita, placed after the factors (§ 112).
Stifel in his Deutsche Arithmetica (Nurnberg, 1545) used the
capital letter M to designate multiplication, and D to designate
division. These letters were again used for this purpose by S. Stevin3
who expresses our Sxyz2 thus: 3 © M sec® M ter @, where sec and ter
mean the "second" and "third" unknown quantities.
The M appears again in an anonymous manuscript of 1638 ex-
plaining Descartes' Geometric of 1637, which was first printed in 1896 ;4
also once in the Introduction to a book by Bartholinus.5
Vieta indicated the product of A and B by writing "A in B"
(Fig. 84). Mere juxtaposition signified multiplication in the Bakhs-
hali tract, in some fifteenth-century manuscripts, and in printed
algebras designating 62 or 5#2; but 5J meant 5+|, not 5X$.
1 H. V. Hilprecht, Babylonian Expedition, Vol. XX, Part 1, Mathematical
etc. Tablets (Philadelphia, 1906), p. 16, 23.
2 T. Eric Peet, The Rhind Mathematical Papyrus (London, 1923), p. 13.
3 (Euvres mathematiques (ed. Albert Girard; Leyden, 1634), Vol. I, p. 7.
4 Printed in (Euvres de Descartes (e"d. Adam et Tannery), Vol. X (Paris,
1908), p. 669, 670.
6 Er. Bartholinus, Renati dea Cartes Principia matheseos universalis (Leyden,
1651), p. 11. See J. Tropfke, op. tit., Vol. II (2d ed., 1921), p. 21, 22.
MULTIPLICATION 251
218. Early uses of the St. Andrew's cross, but not as a symbol of
multiplication of two numbers. — It is well known that the St. Andrew's
cross (X) occurs as the symbol for multiplication in W. Oughtred's
Clavis mathematicae (1631), and also (in the form of the letter X)
in an anonymous Appendix which appeared in E. Wright's 1618 edi-
tion of John Napier's Descriptio. This Appendix is very probably
from the pen of Oughtred. The question has arisen, Is this the earliest
use of X to designate multiplication? It has been answered in the
negative — incorrectly so, we think, as we shall endeavor to show.
In the Encyclopedic des sciences mathematiques, Tome I, Volume
I (1904), page 40, note 158, we read concerning X, "One finds it be-
tween factors of a product, placed one beneath the other, in the Com-
mentary added by Oswald Schreckenfuchs to Ptolemy's Almagest,
1551. "* As will be shown more fully later, this is not a correct inter-
pretation of the symbolism. Not two, but four numbers are involved,
two in a line and two others immediately beneath, thus:
315172^ ,295448
395093/ M74715
The cross does not indicate the product of any two of these numbers,
but each bar of the cross connects two numbers which are multiplied.
One bar indicates the product of 315172 and 174715, the other bar the
product of 395093 and 295448. Each bar is used as a symbol singly;
the two bars are not considered here as one symbol.
Another reference to the use of X before the time of Oughtred is
made by E. Zirkel,2 of Heidelberg, in a brief note in which he protests
against attributing the "invention" of X to Oughtred; he states that
it had a period of development of over one hundred years. Zirkel does
1 Clavdii Ptolemaei Pelusierisis Alexandrini Omnia quae extant Opera (Basileae,
1551), Lib. ii, "Annotationcs."
2Emil Zirkel, Zeitschr. f. math. u. naturw. Vnterricht, Vol. LII (1921), p. 96.
An article on the sign X , which we had not seen before the time of proofreading,
when R. C. Archibald courteously sent it to us, is written by N. L. W. A. Grave-
laar in Wiskundig Tijdschrift, Vol. VI (1909-10), p. 1-25. Gravclaar cites a few
writers whom we do not mention. His claim that, before Oughtred, the sign X
occurred as a sign of multiplication, must be rejected as not borne out by the facts.
It is one thing to look upon X as two symbols, each indicating a separate opera-
tion, and quite another thing to look upon X as only one symbol indicating only
one operation. This remark applies even to the case in § 229, where the four num-
bers involved are conveniently placed at the four ends of the cross, and each
stroke connects two numbers to be subtracted one from the other.
252 A HISTORY OF MATHEMATICAL NOTATIONS
not make his position clear, but if he does not mean that X was
used before Oughtred as a sign of multiplication, his protest is
pointless.
Our own studies have failed to bring to light a clear and conclusive
case where, before Oughtred, X was used as a symbol of multiplica-
tion. In medieval manuscripts and early printed books X was used
as a mathematical sign, or a combination of signs, in eleven or more
different ways, as follows: (1) in solutions of problems by the process
of two false positions, (2) in solving problems in compound proportion
involving integers, (3) in solving problems in simple proportion
involving fractions, (4) in the addition and subtraction of fractions,
(5) in the division of fractions, (6) in checking results of computation
by the processes of casting out the 9's, 7's, or ITs, (7) as part of a
group of lines drawn as guides in the multiplication of one integer by
another, (8) in reducing radicals of different orders to radicals of the
same order, (9) in computing on lines, to mark the line indicating
"thousands," (10) to take the place of the multiplication table above
5 times 5, and (11) in dealing with amicable numbers. We shall
briefly discuss each of these in order.
219. The process of two false positions. — The use of X in this
process is found in the Liber abbaci of Leonardo1 of Pisa, written in
1202. We must begin by explaining Leonardo's use of a single line or
bar. A line connecting two numbers indicates that the two numbers
are to be multiplied together. In one place he solves the problem:
If 100 rotuli are worth 40 libras, how many libras are 5 rotuli worth?
On the margin of the sheet stands the following:
The line connecting 40 and 5 indicates that the two numbers are
to be multiplied together. Their product is divided by 100, but no
symbolism is used to indicate the division, Leonardo uses single lines
over a hundred times in the manner here indicated. In more compli-
cated problems he uses two or more lines, but they do not necessarily
1 Leonardo of Pisa, Liber abbaci (1202) (ed. B. Boncompagni; Roma, 1857),
Vol. I, p. 84.
MULTIPLICATION 253
form crosses. In a problem involving five different denominations of
money he gives the following diagram:1
barcellon. turn. Ian. pisan. imp,
•20 A2^ 13 ^3k 12
11 13 13
barcellon. /* turn. \. Ian. .X^pisan, "X. imp.
11 M2^ 23
Here the answer 20+ is obtained by taking the product of the
connected numbers and dividing it by the product of the unconnected
numbers.
Leonardo uses a cross in solving, by double false position, the
problem: If 100 rotuli cost 13 libras, find the cost of 1 rotulus. The
answer is given in solidi and denarii, where 1 libra = 20 solidi, 1 solidus =
12 denarii. Leonardo assumes at random the tentative answers (the
two false positions) of 3 solidi and 2 solidi. But 3 solidi would
make this cost of 100 rotuli 15 Zi&ra, an error of +2 libras; 2 solidi
would make the cost 10, an error of —3. By the underlying theory of
two false positions, the errors in the answers (i.e., the errors z— 3 and
x — 2 solidi) are proportional to the errors in the cost of 100 rotuli
(i.e., +2 and —3 libras); this proportion yields x = 2 solidi and 7-J
denarii. If the reader will follow out the numerical operations for
determining our x he will understand the following arrangement of the
work given by Leonardo (p. 319) :
uAdditum ex 13 multiplicationibus
4 9
soldi soldi
9
minus
32
Additum ex erroribus.Jt
Observe that Leonardo very skilfully obtains the answer by multiply-
ing each pair of numbers connected by lines, thereby obtaining the
products 4 and 9, which are added in this case, and then dividing 13
by 5 (the sum of the errors). The cross occurring here is not one sym-
bol, but two symbols. Each line singly indicates a multiplication. It
would be a mistake to conclude that the cross is used here as a symbol
expressing multiplication,
i Ibid., Vol. I, p. 127.
254 A HISTORY OF MATHEMATICAL NOTATIONS
The use of two lines crossing each other, in double or single false
position, is found in many authors of later centuries. For example, it
occurs in MS 14908 in the Munich Library,1 written in the interval
1455-64; it is used by the German Widman,2 the Italian Pacioli,3
the Englishman Tonstall,4 the Italian Sfortunati,5 the Englishman
Recorde,6 the German Splenlin,7 the Italians Ghaligai8 and Benedetti,9
the Spaniard Hortega,10 the Frenchman Trenchant,11 the Dutchman
Gemma Frisius,12 the German Clavius,18 the Italian Tartaglia,14 the
Dutchman Sncll,15 the Spaniard Zaragoza,16 the Britishers Jcake17 and
1 See M. Curtze, Zeitschrift f. Math. u. Physik, Vol. XL (Leipzig, 1895).
Supplement, Abhandlungen z. Geschichte d. Mathematik, p. 41.
2 Johann Widman, Behede vnd hubsche Rechenung (Leipzig, 1489). We have
used J. W. L. Glaisher's article in Messenger of Mathematics, Vol. LI (1922), p. 16.
8L. Pacioli, Summa de arithmetica, geometria, etc. (1494). We have used the
1523 edition, printed at Toscolano, fol. 99*, 10O», 182.
4 C. Tonstall, De arte supputandi (1522). We have used the Strassburg edi-
tion of 1544, p. 393.
6 Giovanni Sfortunati da Siena, Nvovo Ijvme. Libro di Arithmetica (1534),
fol. 89-100.
8 R. Recorde, Grovnd of Artes (1543[?]). We have used an edition issued be-
tween 1636 and 1646 (title-page missing), p. 374.
7 Gall Splenlin, Arithmetica kiinstlicher Rechnung (1645). We have used
J. W. L. Glaisher's article in op. cit., Vol. LI (1922), p. 62.
8 Francesco Ghaligai, Pratica d' arithmetica (Nuovamente Rivista ... ; Firenze,
1552), fol. 76.
9 lo. Baptistae Benedicti Divcrsarvm specvlationvm mathematicarum, et physica-
rum Liber (Turin, 1585), p. 105.
10 Juan de Hortega, Tractado subtilissimo de arismetica y de geometria (emenda-
do por Longalo Busto, 1552), fol. 138, 2156.
11 Jan Trenchant, L'arithmetiqve (4th ed.; Lyon, 1578), p. 216.
12 Gemma Frisius, Arithmeticae Practicae methodvs facilis (iam recens ab ipso
authore emcndata .... Parisiis, 1569), fol. 33.
13 Christophori Clavii Bambergensis, Opera mathematica (Mogvntiae, 1612),
Tomus secundus; "Numeratio," p. 58.
14 L'arithmetique de Nicolas Tartaglia Brescian (traduit par Gvillavmo Gosselin
de Caen ... Premier Partie; Paris, 1613), p. 105.
15 Willebrordi Snelli Doctrinae Triangvlorvm Canonicae liber qvatvor (Leyden,
1627), p. 36.
16 Arithmetica Vniversal ... avthor El M. R. P. Joseph Zaragoza (Valencia,
1669), p. 111.
17 Samuel Jeake, AOriSTIKHAOriA or Arithmetick (London, 1696; Preface
1674), p. 501.
MULTIPLICATION 255
Wingate,1 the Italian Guido Grandi,2 the Frenchman Chalosse,3
the Austrian Steinmeyer,4 the Americans Adams5 and Preston.6
As a sample of a seventeenth-century procedure, we give Schott's
solution7 of o~£~~o = 30. He tries z=24 and x =48. He obtains
& u o
errors —25 and —20. The work is arranged as follows:
24.X r48. Dividing 48X25-24X20 by 5
iuf V M gives z= 144.
M. A M. B
25. 5. 20.
220. Compound proportion with integers. — We begin again with
Leonardo of Pisa (1202)8 who gives the problem: If 5 horses eat 6
quarts of barley in 9 days, for how many days will 16 quarts feed 10
horses? His numbers are arranged thus:
The answer is obtained by dividing 9X16X5 by the product of the
remaining known numbers. Answer 12.
Somewhat different applications of lines crossing each other arc
given by Nicolas Chuquet9 and Luca Pacioli10 in dealing with numbers
in continued proportion.
1 Mr. Wingate1 s Arithmetick, enlarged by John Kersey (llth ed.), with supple-
ment by George Shelley (London, 1704), p. 128.
2 Guido Grandi, Instituzioni di arithmelia pratica (Firenze, 1740), p. 104.
1 L'arithmetique par les fractions ... par M. Chalosse (Paris, 1747), p. 158.
4 Tirocinium Arithmeticum a P. Philippo Steinmeyer (Vienna and Freiburg,
1763), p. 475.
6 Daniel Adams, Scholar's Arithmetic (10th ed.; Keene, N.H., 1816), p. 199.
8 John Preston, Lancaster's Theory of Education (Albany, N.Y., 1817), p. 349.
7 G. Schott, Cursus mathematicus (Wtirzburg, 1661), p. 36.
8 Op. cit., p. 132.
• Nicolas Chuquet, Le Triparty en la Science des Nombres (1484), edited by A.
Marre, in BuUettino Boncompagni, Vol. XIII (1880), p. 700; reprint (Roma,
1881), p. 115.
10 Luca Pacioli, op. cit., fol. 93a.
256 A HISTORY OF MATHEMATICAL NOTATIONS
Chuquet finds two mean proportionals between 8 and 27 by the
scheme
" 8 27
3
9
12 18 "
where 12 and 18 are the two mean proportionals sought; i.e., 8, 12, 18,
27 are in continued proportion.
221. Proportions involving fractions. — Lines forming a cross (X),
together with two horizontal parallel lines, were extensively applied
to the solution of proportions involving fractions, and constituted a
most clever device for obtaining the required answer mechanically.
If it is the purpose of mathematics to resolve complicated problems
by a minimum mental effort, then this device takes high rank.
The very earliest arithmetic ever printed, namely, the anonymous
booklet gotten out in 1478 at Treviso,1 in Northern Italy, contains an
interesting problem of two couriers starting from Rome and Venice,
respectively, the Roman reaching Venice in 7 days, the Venetian
arriving at Rome in 9 days. If Rome and Venice are 250 miles apart,
in how many days did they meet, and how far did each travel before
they met? They met in 3}| days. The computation of the distance
traveled by the courier from Rome calls for the solution of the pro-
portion which we write 7:250 = -f-jj- : x.
The Treviso arithmetic gives the following arrangement:
112
.250 63
16
The connecting lines indicate what numbers shall be multiplied to-
gether; namely, 1, 250, and 63, also 7, 1, and 16. The product of the
latter — namely, 112 — is written above on the left. The author then
finds 250X63 = 15,750 and divides this by 112, obtaining 140| miles.
These guiding lines served as Ariadne threads through the maze of
a proportion involving fractions.
We proceed to show that this magical device was used again by
Chuquet (1484), Widman (1489), and Pacioli (1494). Thus Chuquet2
1 The Treviso arithmetic of 1478 is described and partly given in facsimile by
Boncompagni in Atti dell'Accademia Pontificia de' nuovi Lincei, Tome XVI (1862-
63; Roma, 1863), see p. 568.
2 Chuquet, in Boncompagni, Bullettino, Vol. XIII, p. 636; reprint, p. (84).
MULTIPLICATION 257
uses the cross in the problem to find two numbers in the ratio of f
3\ /2
to | and whose sum is 100. He writes - X ~~> multiplying 3 by 3,
4/X3
and 2 by 4, he obtains two numbers in the proper ratio. As their
sum is only 17, he multiplies each by \^ and obtains 47 TV and 52}?.
Johann Widman1 solves the proportion 9 : ^88- = V : x in this man-
9v/53 89
ner: "Secz also - j( — — machss nach der Regel vnd klipt 8 fl.
1/X8 8
35s 9 heir -f$" It will be observed that the computer simply took the
products of the numbers connected by lines. Thus 1X53X89 = 4,717
gives the numerator of the fourth term; 9X8X8 = 576 gives the
denominator. The answer is 8 florins and a fraction.
Such settings of numbers are found in Luca Pacioli,? Ch. Rudolph,3
G. Sfortunati,4 0. Schreckenfuchs,5 Hortega,6 Tartaglia,7 M. Stein-
metz,8 J. Trenchant,9 Hermann Follinus,10 J. Alsted,11 P. H£rigone,12
Chalosse,13 J. Perez de Moya.14 It is remarkable that in England neither
Tonstall nor Recorde used this device. Recorde15 and Leonard Digges16
1 Johann Widman, op. cit.; see J. W. L. Glaisher, op. cit., p. 6.
2 Luca Pacioli, op. cit. (1523), fol. 18, 27, 54, 58, 59, 64.
3 Christoph Rudolph, Kumlliche Rechnung (1526). We have used one of the
Augsburg editions, 1574 or 1588 (title-page missing), CVII.
4 Giovanni Sfortunati da Siena, Nvovo Lvme. Libro di Arithmetica (1534),
fol. 37.
6 O. Schreckenfuchs, op. cit. (1551).
8 Juan de Hortega, op. cit. (1552), fol. 92<z.
7 N. Tartaglia, General Trattato di Nvmeri (la prima partc, 1556), fol. 1116,
117a.
8 Arithmeticae Praecepta . . . . M. Mavricio Steinmetz Gersbachio (Leipzig,
1568) (no paging).
9 J. Trenchant, op. cit., p. 142.
10 Hermann vs Follinvs, Algebra sive liber de rebvs occvltis (Cologne, 1622), p. 72.
11 Johannis-Henrici Alstedii Encyclopaedia (Hernborn, 1630), Lib. XIV,
Cossae libri III, p. 822.
12 Pierre Herigone, Cvrsvs mathematici, Tomus VI (Paris, 1644), p. 320.
18 U Arithmetique par les fractions ... par M. Chalosse (Paris, 1747), p. 71.
14 Juan Perez de Moya, Arithmetica (Madrid, 1784), p. 141. This text reads
the same as the edition that appeared in Salamanca in 1562.
16 Robert Recorde, op. cit., p. 175.
16 (Leonard Digges), A Geometrical Practical Treatise named Pantometria
(London, 1591).
258 A HISTORY OF MATHEMATICAL NOTATIONS
use a slightly different and less suggestive scheme, namely, the capital
letter Z for proportions involving either integers or fractions. Thus,
*) i (*
3 : 8 = 16 : x is given by Recorde in the form Q^^ . This rather un-
usual notation is found much later in the American Accomptant of
Chauncey Lee (Lansinburgh, 1797, p. 223) who writes,
"Cause Effect"
4.5 yds.
and finds Q = 90 X 18 + 4.5 = 360 dollars.
222, Addition and subtraction of fractions. — Perhaps even more
popular than in the solution of proportion involving fractions was the
use of guiding lines crossing each other in the addition and subtrac-
tion of fractions. Chuquet1 represents the addition of f and £ by the
following scheme:
"10 12 "
5
•15-
The lower horizontal line gives 3X5 = 15; we have also 2X5 = 10,
3 X4 = 12; hence the sum f £ = 1-&.
The same line-process is found in Pacioli,2 Rudolph,3 Apianus.4
In England, Tonstall and Recorde do not employ this intersecting
line-system, but Edmund Wingate5 avails himself of it, with only
slight variations in the mode of using it. We find it also in Oronce
Fine,8 Feliciano,7 Schreckenfuchs,8 Hortega,9 Baeza,10 the Italian
1 Nicolas Chuquet, op. tit., Vol. XIII, p. 606; reprint p. (54).
2 Luca Pacioli, op. cit. (1523), fol. 51, 52, 53.
8 Christoph Rudolph, op. cit., under addition and subtraction of fractions.
4 Petrus Apianus, Kauffmansz Rechnung (Ingolstadt, 1527).
6 E. Wingate, op. cit. (1704), p. 152.
6 Orontii Find Delphinatis, liberalivm Disdplinarvm professoris Regii Proto-
mathesis: Opus varium (Paris, 1532), fol. 46.
7 Francesco Feliciano, Libro de arithmetica e geometria (1550).
8 O. Schreckenfuchs, op. cit., "Annot.," fol. 256.
• Hortega, op. cit. (1552), fol. 55a, 636.
10 Nvmerandi doctrina, authore Lodoico Baeza (Paris, 1556), fol. 386.
MULTIPLICATION 259
translation of Fine's works,1 Gemma Frisius,3 Eygaguirre,3 Clavius,4
the French translation of Tartaglia,5 Follinus,6 Girard,7 Hainlin,8
Caramuel,9 Jeake,10 Corachan,11 Chalosse,12 De Moya,13 and in slightly
modified form in Crusoe.14
223. Division of fractions. — Less frequent than in the preceding
processes is the use of lines in the multiplication or division of frac-
tions, which called for only one of the two steps taken in solving a
proportion involving fractions. Pietro Borgi (1488)16 divides f by |
15 "
thus: - x\ - — • In dividing \ by £, Pacioli16 writes
16
"2 3"
and obtains | or 1|.
Petrus Apianus (1527) uses the X in division, Juan de Hortega
(1552) 17 divides f by |, according to the following scheme:
»
, 9
1 Opere di Orontio Fineo del Definato. ... Tradotte da Cosimo Bartoli (Venice,
1587), fol. 31.
2 Arithmeticae Practicae methodvs facilis, per Gemmam Frisium ... iam recens
ab ipso authore emendata ... (Paris, 1569), fol. 20.
3 Sebastian Fernandez Eycaguirre, Libra de Arithmetica (Brussels, 1608), p. 38.
4 Chr. Clavius, Opera omnia, Tom. I (1611), Euclid, p. 383.
6 L' Arithmetique de Nicolas Tartaglia Brescian, traduit ... par Gvillavmo
Gossclin de Caen (Paris, 1613), p. 37.
6 Algebra sive Liber de Rebvs Occvltis, ... Hermann vs Follinvs (Cologne, 1622),
p. 40.
7 Albert Girard, Invention Nouvelle en VAlgebre (Amsterdam, 1629).
8 Johan. Jacob Hainlin, Synopsis malhematica (Tubingen, 1653), p. 32.
8 Joannis Caramvelis Malhesis Biceps Veins et Nova (Companiae, 1670), p. 20.
10 Samuel Jeake, op. cit., p. 51.
11 Juan Bautista Corachan, Arithmetica demonstrada (Barcelona, 1719), p. 87.
w L* Arithmetique par les fractions ... par M. Chalosse (Paris, 17^7), p. 8.
18 J. P. de Moya, op. cit. (1784), p. 103.
14 George E. Crusoe, Y Mathematics? ("Why Mathematics?") (Pittsburgh,
Pa., 1921), p. 21.
15 Pietro Borgi, Arithmetica (Venice, 1488), fol. 33#.
18 L. Pacioli, op. cit. (1523), fol. 54a.
17 Juan de Hortega, op. cit. (1552), fol. 66a.
260 A HISTORY OF MATHEMATICAL NOTATIONS
We find this use of X in division in Sfortunati,1 Blundeville,2
Steinmetz,3 Ludolf van Ceulen,4 De Graaf,5 Samuel Jeake,* and J.
Perez de Moya.7 De la Chapelle, in his list of symbols,8 introduces
X as a regular sign of division, divisG par, and x as a regular sign of
multiplication, multipli6 par. He employs the latter regularly in
multiplication, but he uses the former only in the division of fractions,
and he explains that in |Xf = $f, "le sautoir X montre que 4 doit
multiplier 6 & que 3 doit multiplier 7," thus really looking upon X
as two symbols, one placed upon the other.
224. In the multiplication of fractions Apianus9 in 1527 uses the
1 o
parallel horizontal lines, thus, ~ •= . Likewise, Michael Stifel10 uses
A O
two horizontal lines to indicate the steps. He says: "Multiplica
numeratores inter se, et proveniet numerator productac summae.
Multiplica etiam denominatores inter se, et proveniet denominator
productae summae."
225. Casting out the 9's, 7's, or ll's. — Checking results by casting
out the 9's was far more common in old arithmetics than by casting
out the 7's or 1 1's. Two intersecting lines afforded a convenient group-
ing of the four results of an operation. Sometimes the lines appear in
the form X, at other times in the form +• Luca Pacioli11 divides
97535399 by 9876, and obtains the quotient 9876 and remainder 23.
Casting out the 7's (i.e., dividing a number by 7 and noting the
residue), he obtains for 9876 the residue 6, for 97535399 the residue 3,
"62 "
for 23 the residue 2. He arranges these residues thus: >, Q- .
u o
Observe that multiplying the residues of the divisor and quotient,
6 times 6 = 36, one obtains 1 as the residue of 36. Moreover, 3—2
is also 1. This completes the check.
1 Giovanni Sfortvnati da Siena, Nvovo Lvme. Libra di Arithmetica (1534),
fol. 26.
2 Mr. Blundevil. His Exercises conlayning eight Treatises (London, 1636), p. 29.
3 M. Mavricio Steinmetz Gersbachio, Arithmetical praecepta (1568) (no paging).
4 Ludolf van Ceulen, De arithm. (title-page gone) (1615), p. 13.
5 Abraham de Graaf, De Geheele Mathesis of Wiskonst (Amsterdam, 1694),
p. 14.
9 Samuel Jeake, op. cU.t p. 58. 7 Juan Perez de Moya, op. cit., p. 117.
8 De la Chapelle, Institutions de geometric (4th eU; Paris, 1765), Vol. I, p. 44,
118, 185.
* Petrus Apianus, op. cit. (1527).
10 M. Stifel, Arithmetica integra (Nuremberg, 1544), fol. 6.
11 Luca Pacioli, op. cit. (1523), fol. 35.
MULTIPLICATION 261
Nicolas Tartaglia1 checks, by casting out the 7's, the division
912345 + 1987 = 459 and remainder 312.
Casting the 7's out of 912345 gives 0, out of 1987 gives 6,
" 4'4 "
out of 459 gives 4, out of 312 gives 4. Tartaglia writes down -j- .
OiLF
Here 4 times 6 = 24 yields the residue 3; 0 minus 4, or better 7
minus 4, yields 3 also. The result "checks."
Would it be reasonable to infer that the two perpendicular lines +
signified multiplication? We answer "No," for, in the first place, the
authors do not state that they attached this meaning to the symbols
and, in the second place, such a specialized interpretation does not
apply to the other two residues in each example, which are to be
subtracted one from the other. The more general interpretation, that
the lines are used merely for the convenient grouping of the four resi-
dues, fits the case exactly.
Rudolph2 checks the multiplication 5678 times 65 = 369070 by
casting out the 9's (i.e., dividing the sum of the digits by 9 and noting
the residue); he finds the residue for the product to be 7, for the
factors to be 2 and 8. He writes down
Here 8 times 2 = 16, yielding the residue 7, written above. This
residue is the same as the residue of the product; hence the check is
complete. It has been argued that in cases like this Rudolph used X
to indicate multiplication. This interpretation does not apply to
other cases found in Rudolph's book (like the one which follows) and
is wholly indefensible. We have previously seen that Rudolph used
X in the addition and subtraction of fractions. Rudolph checks the
proportion 9: 11 =48: x, where x = 58|, by casting out the 7's, 9's,
and iTs as follows:
"(7) (9) (II)'1
\6/ vO, 0)/
Take the check by ITs (i.e., division of a number by 11 and noting
the residue). It is to be established that 9# = 48 times 11, or that 9
1 N. Tartaglia, op. cit. (1556), fol. 34£.
a Chr. Rudolph, Kunstliche Rechnung (Augsburg, 1574 or 1588 ed.) A VIII.
262 A HISTORY OF MATHEMATICAL NOTATIONS
times 528 = 48 times 99. Begin by casting out the ll's of the factors 9
and 48; write down the residues 9 and 4. But the residues of 528 and
99 are both 0. Multiplying the residues 9 and 0, 4 and 0, we obtain
in each case the product 0. This is shown in the figure. Note that here
we do not take the product 9 times 4; hence X could not possibly in-
dicate 9 times 4.
The use of X in casting out the 9's is found also in Recorders
Grovnd of Aries and in Clavius1 who casts out the 9's and also the 7's.
Hortega2 follows the Italian practice of using lines +, instead
of X, for the assignment of resting places for the four residues con-
sidered. Hunt3 uses the Latin cross — |- . The regular X is used by
Regius (who also casts out the 7's),4 Lucas,5 Metius,6 Alsted,7 York,8
Dechales,9 Ayres,10 and Workman.11
In the more recent centuries the use of a cross in the process of
casting out the 9?s has been abandoned almost universally; we have
found it given, however, in an English mathematical dictionary12 of
1814 and in a twentieth -century Portuguese cyclopedia.13
226. Multiplication of integers. — In Pacioli the square of 37 is
found mentally with the aid of lines indicating the digits to be multi-
plied together, thus:
<
1369
^hr. Clavius, Opera omnia (1612), Tom. I (1611), "Numeratio," p. 11.
2 Juan de Hortega, op. tit., fol. 426.
3 Nicolas Hunt, Hand-Maid to Arithmetick (London 1633).
4 Hudalrich Regius, Vtrivsgve Arithmetices Epitome (Strasburg, 1536), fol. 57;
ibid. (Freiburg-in-Breisgau, 1543), fol. 56.
6Lossius Lucas, Arithmelices Erotemata Pverilia (Liirieburg, 1569), fol. 8.
6 Adriani Metii Alcmariani Arithmeticae libri dvo: Leyden, Arith. Liber I,
P. 11.
7Johann Heinrich Alsted, Methodus Admirandorum mathcmaticorum novem
Libris (Tertia editio; Herbon, 1641), p. 32.
<Tho. York, Practical Treatise of Arithmetick (London, 1687), p. 38.
9R. P. Claudii Francisci Milliet Dechales Camberiensis, Mundus Mathe-
maticus. Tomus Primus, Editio altera (Leyden, 1690), p. 369.
10 John Ayres, Arithmetick made Easie, by E. Hatton (London, 1730), p. 53.
11 Benjamin Workman, American Accountant (Philadelphia, 1789), p. 25.
12 Peter Barlow, Math. <fc Phil. Dictionary (London, 1814), art. "Multiplica-
tion."
13 Encyclopedia Portugueza (Porto), art. "Nove."
MULTIPLICATION 263
From the lower 7 two lines radiate, indicating 7 times 7, and 7 times 3,
Similarly for the lower 3. We have here a cross as part of the line-
complex. In squaring 456 a similar scheme is followed ; from each digit
there radiate in this case three lines. The line-complex involves three
vertical lines and three well-formed crosses X. The multiplication
of 54 by 23 is explained in the manner of Pacioli by Mario Bettini1
in 1642.
There are cases on record where the vertical lines are omitted,
either as deemed superfluous or as the result of an imperfection in the
typesetting. Thus an Italian writer, Unicorno,2 writes:
"7 8"
x
4368
It would be a rash procedure to claim that we have here a use of
X to indicate the product of two numbers; these lines indicate the
product of 6 and 70, and of 50 and 8; the lines are not to be taken as
one symbol; they do not mean 78 times 56. The capital letter X is
used by F. Ghaligai in a similar manner in his Algebra. The same re-
marks apply to J. H. Alsted3 who uses the X, but omits the vertical
lines, in finding the square of 32.
A procedure resembling that of Pacioli, but with the lines marked
as arrows, is found in a recent text by G. E. Crusoe.4
227. Reducing radicals to radicals of the same order. — Michael
Stifel6 in 1544 writes: "Vt volo reducere \/z 5 et j/c£ 4 ad idem signum,
sic stabit exemplum ad regulam
5 4
x
V* V4.
1 Mario Bettino, Apiaria Vniversae philosophiae mathematicae (Bologna,
1642), "Apiarivm vndecimvm," p. 37.
2S. Joseppo Vnicorno, De Varithmetica universale (Venetia, 1598), fol. 20.
Quoted from C. le Paige, "Sur 1'origine de certains signes d'opfaation," Anncdes
de la soctitt scientifiyue de Bruxelles (16th year, 1891-92), Part II, p. 82.
8J. H. Alsted, Methodus Admirandorum Maihematicorum Novem libris e&-
hibens universam mathesin (tertiam editio; Herbon, 1641), p. 70.
4 George E. Crusoe, op. ct/., p. 6.
6 Michael Stifel, Arithmetica integra (1544), fol. 114.
264 A HISTORY OF MATHEMATICAL NOTATIONS
j/zc£125 et j/2c£16," Here j/5 and ^4 are reduced to radicals of the
same order by the use of the cross X. The orders of the given radicals
are two and three, respectively ; these orders suggest the cube of 5 or
125 and the square of 4, or 16. The answer is 1/125 and 1/16.
Similar examples are given by Stifel in his edition of RudolfTs
Cossyl Peletier,2 and by De Billy.3
228. To mark the place for "thousands." — In old arithmetics
explaining the computation upon lines (a modified abacus mode of
computation), the line on which a dot signified "one thousand" was
marked with a X. The plan is as follows:
X - - 1000
500
- —100
50
- 50
5
- 1
This notation was widely used in Continental and English texts.
229. In place of multiplication table above 5X5. — This old pro-
cedure is graphically given in Recorders Grovnd of Artes (1543?). Re-
quired to multiply 7 by 8. Write the 7 and 8 at the cross as shown
here; next, 10 — 8 = 2, 10 — 7, = 3; write the 2 and 3 as shown:
"8 2"
56
Then, 2X3 = 6, write the 6; 7-2 = 5, write the 5. The required
product is 56. We find this process again in Oronce Fine,4 Regius,6
1 Michael Stifel, Die Coss Christoffs Rudolfs (Amsterdam, 1615), p. 136.
(First edition, 1553.)
2 Jacobi Peletarii Cenomani, de occvlta parte nvmerorvm, qvam Algebram vacant,
Libri duo (Paris, 1560), fol. 52.
s Jacqves de Billy, Abregt des Preceptes d'Algebre (Reims, 1637), p. 22. See
also the Nova Geometriae Clavis, authore P. Jacobo de Billy (Paris, 1643), p. 465.
4 Orontii Finei Delphinatis, liberalivm Disciplinarvm prefossoris Regii Proto-
mathesis: Opus uarium (Paris, 1532), fol. 4b.
6 Hudalrich Regius, Vtrivsqve arithmetices Epitome (Strasburg, 1536), fol. 53;
ibid. (Freiburg-in-Breisgau, 1543), fol. 56.
MULTIPLICATION
265
Stifel,1 Boissiere,2 Lucas,3 the Italian translation of Oronce Fine,4
the French translation of Tartaglia,5 Alsted,8 Bettini.7 The French
edition of Tartaglia gives an interesting extension of this process,
which is exhibited in the product of 996 and 998, as follows:
996
994 0 0 8
230. Amicable numbers. — N. Chuquet8 shows graphically that
220 and 284 are amicable numbers (each the sum of the factors of the
other) thus:
"220 /2S4"
110\/ 142
4
2
1
220
44
22
20
11
10
5
4
2
1
284
The old graphic aids to computation which we have described are
interesting as indicating the emphasis that was placed by early arith-
meticians upon devices that appealed to the eye and thereby con-
tributed to economy of mental effort.
231. The St. Andrew1 s cross used as a symbol of multiplication. —
As already pointed out, Oughtred was the first (§ 181) to use X as the
1 Michael Stifel, Arithmetica integra (Nuremberg, 1544), fol. 3.
2 Claude de Boissiere, Daulphinois, UArtd'Arythmetique (Paris, 1554), fol. 156.
3 Lossius Lucas, Arithmetices Erotemata Pverilia (Liineburg, 1569), fol. 8.
4 Opere di Orontio Fineo. ... Tradotte da Cosimo Bartoli (Bologna, 1587),
"Delia arismetica," libro primo, fol. 6, 7.
6 L'arithmetique de Nicolas Tartaglia ... traduit ... par Gvillavmo Gosselin de
Caen. (Paris, 1613), p. 14.
6 Johannis-Henrici Alstedii Encyclopaedia (Herbon, 1630), Lib. XIV, p. 810.
* Mario Bettino, Apiaria (Bologna, 1642), p. 30, 31.
8 N. Chuquet, op. tit., VoL XIII, p. 621; reprint, p. (69).
266 A HISTORY OF MATHEMATICAL NOTATIONS
sign of multiplication of two numbers, as aXb (see also §§ 186, 288).
The cross appears in Oughtred's Clavis mathematicae of 1631 and, in
the form of the letter X, in E. Wright's edition of Napier's Descriptio
(1618). Oughtred used a small symbol X for multiplication (much
smaller than the signs + and — ). In this practice he was followed
by some writers, for instance, by Joseph Moxon in his Mathematical
Dictionary (London, 1701), p. 190. It seems that some objection had
been made to the use of this sign X, for Wallis writes in a letter of
September 8, 1668: "I do not understand why the sign of multi-
plication X should more trouble the convenient placing of the frac-
tions than the older signs + — = >:: ."* It may be noted that
Oughtred wrote the X small and placed it high, between the factors.
This practice was followed strictly by Edward Wells.2
On the other hand, in A. M. Legendre's famous textbook Geometric
(1794) one finds (p. 121) a conspicuously large-sized symbol X> f°r
multiplication. The following combination of signs was suggested by
Stringham:3 Since X means "multiplied by," and / "divided by,"
the union of the two, viz., X/, means "multiplied or divided by."
232. Unsuccessful symbols for multiplication. — In the seventeenth
century a number of other designations of multiplication were pro-
posed. H£rigone4 used a rectangle to designate the product of two
factors that were separated by a comma. Thus, "D5+4+3, 7~3:
~10,es«38" meant (5+4+3)- (7-3)-10 = 38. Jones, in his Synopsis
palmariorum (1706), page 252, uses the us, the Hebrew letter mem, to
denote a rectangular area. A six-pointed star was used by Rahn arid,
after him, by Brancker, in his translation of Rahn's Teutsche Algebra
(1659). "The Sign of Multiplication is [-#] i.e., multiplied with."
We encounter this use of -X- in the Philosophical Transactions.5
Abraham de Graaf followed a practice, quite common among
Dutch writers of the seventeenth and eighteenth centuries, of placing
symbols on the right of an expression to signify direct operations
(multiplication, involution), and placing the same symbols on the
1 S. P. Rigaud, Correspondence of Scientific Men of the Seventeenth Century
(Oxford, 1841), Vol. II, p. 494.
* Edward Wells, The Young Gentleman's Arithmetic and Geometry <2d ed.;
London, 1723); "Arithmetic," p. 16, 41; "Geometry," p. 283, 291.
3 Irving Stringham, Uniplanar Algebra (San Francisco, 1893), p. xiii.
4 P. Herigone, Cursus mathematici (1644), Vol. VI, explicatio notarum.
(First edition, 1642.)
5 Philosophical Transactions, Vol. XVII, (1692-94), p. 680. See also §§ 194,
547.
MULTIPLICATION 267
left of an expression to signify inverse operations. Thus, Graaf1
multiplies x*+4 by 2J by using the following symbolism:
, " xx tot 4 . "
als f xx tot 9
In another place he uses this same device along with double commas,
thus
<t — 7— r »
a+b , , — cc ,
a+b , , — ccd
to represent (a+b)( — cc) (d) = (a + b) ( — ccd) .
Occasionally the comma was employed to mark multiplication, as
23 23 112
in Herigone (§ 189), F. Van Schooten,2 who in 1657 gives — — — '-— ~ •,
Q *2 '2 i /I 1 Q r
3,0,0 I/ 11O,0
where all the commas signify "times/' as in Leibniz (§§ 197, 198, 547),
in De Gua3 who writes "3, 4, 5 , ... &c. n— w— "2," in Petrus Hor-
rebowius4 who lets "AtB" stand for A times B, in Abraham de Graaf5
who uses one or two commas, as in "p — 6,a" for (p — 6)a. The German
Htibsch6 designated multiplication by /-, as in l/^.
233. The dot for multiplication. — The dot was introduced as a
symbol for multiplication by G. W. Leibniz. On July 29, 1098, ho
wrote in a letter to John Bernoulli: "I do not like X as a symbol for
multiplication, as it is easily confounded with a:; .... often I simply
relate two quantities by an interposed dot and indicate multiplication
by ZC - LM. Hence, in designating ratio I use not one point but two
poiiits, which I use at the same time for division." It has been stated
that the dot was used as a symbol for multiplication before Leibniz,
that Thomas Harriot, in his Artis analyticae praxis (1631), used the
dot in the expressions "aaa— 3-W>a=+2-ccc." Similarly, in explain-
ing cube root, Thomas Gibson7 writes, in 1655, "3-66," "3 -6cc," but it
1 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 8.
2 Francisci ct Schooten. ... Exercitationum mathematicarum liber primus
(Leyden, 1657), p. 89.
3 L'Abbe' de Gua, Histoire de I'academie r. d. sciences, annee 1741 (Paris,
1744), p. 81. '
4 Pelri Horrebowii Operum mathematico-physicorum tomus primus
(Havniae, 1740), p. 4.
5 Abraham de Graaf, op. cit. (1672), p. 87.
« J. G. G. Htibsch, Arithmetica Portensis (Leipzig, 1748). Taken from Wilder-
muth's article, "Rechnen," in K. A. Schmid;s Encyklopaedie des gesammten Er-
ziehungs- und Unterrichtswesens (1885).
7 Tho. Gibson, Syntaxis mathematica (London, 1655), p. 36.
268 A HISTORY OF MATHEMATICAL NOTATIONS
is doubtful whether either Harriot or Gibson meant these dots for
multiplication. They are introduced without explanation. It is much
more probable that these dots, which were placed after numerical
coefficients, are survivals of the dots habitually used in old manu-
scripts and in early printed books to separate or mark off numbers
appearing in the running text. Leibniz proposed the dot after he had
used other symbols for over thirty years. In his first mathematical
publication, the De arte combinatoria1 of 1666, he used a capital letter
C placed in the position O for multiplication, and placed in the
position O for division. We have seen that in 1698 he advocated the
point. In 1710 the Leibnizian symbols2 were explained in the publica-
tion of the Berlin Academy (§ 198); multiplication is designated by
apposition, and by a dot or comma (punctum vel comma), as in 3,2 or
a,b+c or AB,CD+EF. If at any time some additional symbol is de-
sired, O is declared to be preferable to X .
The general adoption of the dot for multiplication in Europe in the
eighteenth century is due largely to Christian Wolf. It was thus used
by L. Euler; it was used by James Stirling in Great Britain, where the
Oughtredian X was very popular.3 Whit worth4 stipulates, "The
full point is used for the sign of multiplication."
234. The St. Andrew's cross in notation for transjinite ordinal
numbers. — The notation o>X2, with the multiplicand on the left, was
chosen by G. Cantor in the place of 2co (where w is the first transfinite
ordinal number), because in the case of three ordinal transfinite
numbers, a, (I, 7, the product a? • a7 is equal to of^ when oP is the
multiplicand, but when ay is the multiplicand the product is a7^. In
transfinite ordinals, £+7 is not equal
SIGNS FOR DIVISION AND RATIO
235. Early symbols. — Hilprecht5 states that the Babylonians
had an ideogram IGI-GAL for the expression of division. Aside from
their fractional notation (§ 104), the Greeks had no sign for division.
Diophantus6 separates the dividend from the divisor by the words Iv
1 G. W. Leibniz, Opera omnia, Vol. II (Geneva, 1768), p. 347.
2 Miscellanea Berolinensia (Berlin), Vol. I (1710), p. 156.
' See also §§ 188, 287, 288; Vol. II, §§ 541, 547.
4 W. A. Whitworth, Choice and Chance (Cambridge, 1886), p. 19.
6H. V. Hilprecht, The Babylonian Expedition Mathematical, etc., Tablets from
the Temple Library of Nippur (Philadelphia, 1906), p. 22.
6 Diophantus, Arithmetica (ed. P. Tannery; Leipzig, 1893), p. 286. See also
G. H. F. Nesselmann, Algebra der Griechen (Berlin, 1842), p. 299.
DIVISION AND RATIO 269
or noplov, as in the expression 6VJ \d\f/a ss ic6 /ioptou 6°a/xd IJ3
f, which means (7z2 — 24x)-f-(x2+12— 7z). In the Bakhshali
arithmetic (§ 109) division is marked by the abbreviation bhd from
bhdga, "part." The Hindus often simply wrote the divisor beneath
the dividend. Similarly, they designated fractions by writing the
denominator beneath the numerator (§§ 106, 109, 113). The Arabic
author1 al-Ha?$ar, who belongs to the twelfth century, mentions the
use of a fractional line in giving the direction: " Write the denomina-
tors below a [horizontal] line and over each of them the parts belonging
to it; for example, if you are told to write three-fifths and a third of a
3 1
fifth, write thus, H~O«" ^n a second example, four-thirteenths and
O o
4 3
three-elevenths of a thirteenth is written ^ — ^. This is the first
lo 11
appearance of the fractional line, known to us, unless indeed Leonardo
of Pisa antedates al-Ha$sar. That the latter was influenced in this
matter by Arabic authors is highly probable. In his Liber abbaci
(1202) he uses the fractional line (§ 122). Under the caption2 "De
diuisionibus integrorum numcrorum" he says: "Cum super quem-
libet numerum quedam uirgula protracta fuerit, et super ipsam qui-
libet alius numerus descriptus fuerit, superior numerus partem uel
partes inferioris numeri affirmat; nam inferior denominatus, et su-
perior denominans appellatur. Vt si super binarium protracta fuerit
uirgula, et super ipsam unitas descripta sit ipsa unitas unam part-cm
de duabus partibus unius integri affirmat, hoc est medietatem sic |."
("When above any number a line is drawn, and above that is written
any other number, the superior number stands for the part or parts
of the inferior number; the inferior is called the denominator, the
superior the numerator. Thus, if above the two a line is drawn,
and above that unity is written, this unity stands for one part of two
parts of an integer, i.e., for a half, thus £.") With Leonardo, an indi-
cated division and a fraction stand in close relation. Leonardo writes
157
also . o"r"Yri> w^ich means, as he explains, seven-tenths, and five-
Zi U lU
sixths of one-tenth, and one-half of one-sixth of one-tenth.
236. One or two lunar signs, as in 8)24 or 8)24(, which are often
employed in performing long and short division, may be looked upon
as symbolisms for division. The arrangement 8)24 is found in Stifel's
1 H. Sutcr, Bibliotheca mathcmatica (3d scr.), Vol. II (1901), p. 24.
2 // Liber abbaci di Leonardo Pisano (ed. B. Boncompagni; Roma, 1857),
p. 23, 24.
270 A HISTORY OF MATHEMATICAL NOTATIONS
Arithmetica Integra (1544)1, and in W. Oughtred's different editions of
his Claris mathematicae. In Oughtred's Opuscula posthuma one finds
also $]|[f, (§182). Joseph Moxon2 lets D)A+B-C signify our
Perhaps the earliest to suggest a special symbol for division other
than the fractional line, and the arrangement 5)15 in the process of
dividing, was Michael Stifel3 in his Deutsche Arithmetica (1545). By
the side of the symbols + and — he places the German capitals 2ft
and 35, to signify multiplication and division, respectively. Strange
to say, he did not carry out his own suggestion; neither he nor seem-
ingly any of his German followers used the 50} and SD in arithmetic or
algebraic manipulation. The letters M and D are found again in S.
5x"
Stevin, who expressed our -- - • z2 in this manner.'4
y
5®D sec ®M ter ® ,
where sec and ter signify the "second" and "third" unknown quantity.
The inverted letter Q is used to indicate division by Gallimard,6
as in
"12 Q 4 = 3" and "a262 <I a2."
In 1790 Da Cunha6 uses the horizontal letter TJ as a mark for division.
237. Rahris notation. — In 1659 the Swiss Johann Heinrich Rahn
published an algebra7 in which he introduced -r- as a sign of division
(§ 194). Many writers before him had used ^ as a minus sign (§§ 164,
208). Rahn's book was translated into English by Thomas Brancker
(a graduate of Exeter College, Oxford) and published, with additions
from the pen of Joh. Pell, at London in 1668. Rahn's Teutsche Algebra
was praised by Leibniz8 as an "elegant algebra," nevertheless it did
not enjoy popularity in Switzerland and the symbol -f- for division
1 Michael Stifel, Arithmetica Integra (Nurnberg, 1544), fol. 317F°, 318r°.
This reference is taken from J. Tropfke, op. tit., Vol. II (2d ed., 1921 ), p. 28, n. 114.
2 Joseph Moxon, Mathematical Dictionary (3ded.; London, 1701), p. 190, 191.
3 Michael Stifel, Deutsche Arithmetica (Nurnberg, 1545), fol. 74^°. We draw
this information from J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 21.
4 S. Stevin, (Euvres (ed. A. Girard, 1634), Vol. I, p. 7, def . 28.
6 J. E. Gallimard, La Science du calcul numerique, Vol. I (Paris, 1751), p. 4;
Methvde ... d'arithmetique, d'algebre et de geometric (Paris, 1753), p. 32.
6 J. A. da Cunha, Principles mathematicos (1790), p. 214.
7 J. H. Rahn, Teutsche Algebra (Zurich, 1659).
8 Leibnizens mathematische Schriften (ed. C. I. Gerhardt), Vol. VII, p. 214.
DIVISION AND RATIO 271
was not adopted by his countrymen. In England, the course of events
was different. The translation met with a favorable reception;
Rahn's -f- and some other symbols were adopted by later English
writers, and came to be attributed, not to Rahn, but to John Pell. It
so happened that Rahn had met Pell in Switzerland, and had received
from him (as Rahn informs us) the device in the solution of equations
of dividing the page into three columns and registering the successive
steps in the solution. Pell and Brancker never claimed for themselves
the introduction of the -f- and the other symbols occurring in Rahn's
book of 1569. But John Collins got the impression that not only the
three-column arrangement of the page, but all the new algebraic
symbols were due to Pell. In his extensive correspondence with
John Wallis, Isaac Barrow, and others, Collins repeatedly spoke of -f-
as "Pell's symbol." There is no evidence to support this claim (§ 194) .1
. The sign -r- as a symbol for division was adopted by John Wallis
and other English writers. It came to be adopted regularly in Great
Britain and the United States, but not on the European Continent. The
only text not in the English language, known to us as using it, is one
published in Buenos Aires;2 where it is given also in the modified form
•/•, as in f •/• 8=f&. In an American arithmetic,3 the abbreviation
-i-rs was introduced for "divisors," and -s-nds for "dividends," but
this suggestion met with no favor on the part of other writers.
238. Leibniz' notations. — In the Dissertatio de arte combinatoria
(1668)4 G. W. Leibniz proposed for division the letter C, placed hori-
zontally, thus O, but he himself abandoned this notation and in-
troduced the colon. His article of 1684 in the Ada eruditorum
contains for the first time in print the colon ( : ) as the symbol for di-
vision.5 Leibniz says: ". . . . notetur, me divisionem hie designare hoc
/r
modo: x:y, quod idem est ac x divis. per y seu - ." In a publication of
a
the year 17108 we read: "According to common practice, the division
1 F. Cajori, "Rahn's Algebraic Symbols," Amer. Math. Monthly, Vol. XXXI
(1924), p. 65-71.
2 Florentine Garcia, El aritme'tico Argentina (5th ed.; Buenos Aires, 1871),
p. 102. The symbol ~ and its modified form are found in the first edition of this
book, which appeared in 1833.
8 The Columbian Arithmetician, "by an American'1 (Haverhill [Mass.], 1811),
p. 41.
4 Leibniz, Opera omnia, Tom. II (Geneva, 1768), p. 347.
6 See Leibnizew mathematische Schriften (ed. C. I. Gerhardt), Vol. V (1858),
p. 223. See also M. Cantor, Gesch. d. Mathematik, Vol. Ill (2d ed.; Leipzig), p. 194.
6 Miscellanea Berolinensia (Berlin, 1710), p. 156. See our § 198.
272 A HISTORY OF MATHEMATICAL NOTATIONS
is sometimes indicated by writing the divisor beneath the dividend,
with a line between them; thus a divided by 6 is commonly indicated
by r/'Very often however it is desirable to avoid this and to continue
on the same line, but with the interposition of two points; so that a:b
means a divided by b. But if, in the next place a : b is to be divided by
c, one may write a : 6, : c, or (a : b) : c. Frankly, however, in this case the
relation can be easily expressed in a different manner, namely a: (be)
or a:bc, for the division cannot always be actually carried out but
often can only be indicated and then it becomes necessary to mark the
course of the deferred operation by commas or parentheses."
In Germany, Christian Wolf was influential through his textbooks
in spreading the use of the colon (:) for division and the dot (•) for
multiplication. His influence extended outside Germany. A French
translation of his text1 uses the colon for division, as in "(a—b):b."
He writes: "a:mac = b:mbc."
239. In Continental Europe the Leibnizian colon has been used
for division and also for ratio. This symbolism has been adopted in
the Latin countries with only few exceptions. In 1878 Balbontin2
used in place of it the sign -f- preferred by the English-speaking
countries. Another Latin-American writer3 used a slanting line in
this manner, (y\3 1 =-^-- = = and also 12\3 = 4. An author in Peru4
indicates division by writing the dividend and divisor on the same
line, but inclosing the former in a parenthesis. Accordingly, "(20)5"
meant 20-^5. Sometimes he uses brackets and writes the proportion
2:11 = 20:15 in this manner: "2:1[1J2: :20:15."
240. There are perhaps no symbols which are as completely ob-
servant of political boundaries as are -f- and : as symbols for division.
The former belongs to Great Britain, the British dominions, and the
United States. The latter belongs to Continental Europe and the
Latin-American countries. There are occasional authors whose prac-
1 C. Wolf, Cours de mathematique, Tom. I (Paris, 1747), p. 110, 118.
2 Juan Maria Balbontin, Tratado elemental de arilmetica (Mexico, 1878), p. 13.
3 Felipe Senillosa, Tratado elemental de ariftmetica (neuva ed.; Buenos Aires,
1844), p. 16. We quote from p. 47: "Este signo deque hemos hecho uso en la
partition (\) no es usado generalmente; siendo el que se usa los dos punctos (:)
6 la forma de quebrado. Pero un quebrado denota mas bien un cociente 6 particion
ejecutada que la operacion 6 acto del partir; asf hemos empleado este signo \ con
analogia al del multiplicar que es e*ste: X."
4 Juan de Dios Salazar, Lecciones de aritmetica (Arequipa, 1827), p. v, 74, 89.
DIVISION AND RATIO 273
tices present exceptions to this general statement of boundaries, but
their number is surprisingly small. Such statements would not apply
to the symbolisms for the differential and integral calculus, not even
for the eighteenth century. Such statements would not apply to
trigonometric notations, or to the use of parentheses or to the desig-
nation of ratio and proportion, or to the signs used in geometry.
Many mathematical symbols approach somewhat to the position
of world-symbols, and approximate to the rank of a mathematical
world-language. To this general tendency the two signs of division
-r- and : mark a striking exception. The only appearance of -f- signi-
fying division that we have seen on the European Continent is in an
occasional translation of an English text, such as Colin Maclaurin's
Treatise of Algebra which was brought out in French at Paris in 1753.
Similarly, the only appearance of : as a sign for division that we have
seen in Great Britain is in a book of 1852 by T. P. Kirkman.1 Saverien2
argues against the use of more than one symbol to mark a given
operation. "What is more useless and better calculated to disgust a
beginner and embarrass even a geometer than the three expressions
-, :, -f-, to mark division?"
241. Relative position of divisor and dividend. — In performing the
operation of division, the divisor and quotient have been assigned
various positions relative to the dividend. When the "scratch
method" of division was practiced, the divisor was placed beneath
the dividend and moved one step to the right every time a new figure
of the quotient was to be obtained. In such cases the quotient was
usually placed immediately to the right of the dividend, but some-
times, in early writers, it was placed above the dividend. In short
division, the divisor was often placed to the left of the dividend, so
that a)b(c came to signify division.
A curious practice was followed in the Dutch journal, the Maan-
delykse Mathematische Liefhebberye (Vol. I [1759], p. 7), where a)
signifies division by a, and (a means multiplication by a. Thus:
** />»-)/ — 7) /7_1_7» "
M\ xy — u a-f.c
_b—a+x *
~~ x
James Thomson called attention to the French practice of writing
the divisor on the right. He remarks: "The French place the divisor
1 T. P. Kirkman, First Mnemonial Lessons in Geometry, Algebra and Trigo-
nometry (London, 1852).
2Alexandre Saverien, Dictionnaire universel de maihematique et de physique
(Paris, 1753), "Caractere."
274 A HISTORY OF MATHEMATICAL NOTATIONS
to the right of the dividend, and the quotient below it This
mode gives the work a more compact and neat appearance, and pos-
sesses the advantage of having the figures of the quotient near the
divisor, by which means the practical difficulty of multiplying the
divisor by a figure placed at a distance from it is removed
This method might, with much propriety, be adopted in preference
to that which is employed in this country."1
The arrangement just described is given in B£zout's arithmetic,2
in the division of 14464 by 8, as follows:
"144641 8_ „
|i808 '
242. Order of operations in terms containing both -f- and X . — If an
arithmetical or algebraical term contains -f- and X , there is at present
no agreement as to which sign shall be used first. "It is best to avoid
such expressions. "3 For instance, if in 24-7-4X2 the signs are used as
they occur in the order from left to right, the answer is 12; if the sign
X is used first, the answer is 3.
Some authors follow the rule that the multiplications and divi-
sions shall be taken in the order in which they occur.4 Other textbook
writers direct that multiplications in any order be performed first,
then divisions as they occur from left to right.5 The term a~bXb is
interpreted by Fisher and Schwatt6 as (a-r6)X&. An English com-
mittee7 recommends the use of brackets to avoid ambiguity in such
cases.
243. Critical estimates of : and -r- as symbols. — D. Andre8 expresses
himself as follows: "The sign : is a survival of old mathematical no-
tations; it is short and neat, but it has the fault of being symmetrical
toward the right and toward the left, that is, of being a symmetrical
sign of an operation that is asymmetrical. It is used less and less.
1 James Thomson, Treatise on Arithmetic (18th ed.; Belfast, 1837).
2 Arithmetique de Bezout ... par F. Peyrard (13th ed.; Paris, 1833).
3 M. A. Bailey, American Menial Arithmetic (New York, 1892), p. 41.
4 Hawkes, Luby, and Teuton, First Course of Algebra (New York, 1910), p. 10.
5 Slaught and Lennes, High School Algebra, Elementary Course (Boston, 1907),
p. 212.
6 G. E. Fisher and I. J. Schwatt, Text-Book of Algebra (Philadelphia, 1898),
p. 85.
7 "The Report of the Committee on the Teaching of Arithmetic in Public
Schools," Mathematical Gazette, Vol. VIII (1917), p. 238. See also p. 290.
8 Desire* Andre*, Des Notations mathtmatiques (Paris, 1909), p. 58, 59.
DIVISION AND RATIO 275
.... When it is required to write the quotient of a divided by b, in
the body of a statement in ordinary language, the expression a: b
really offers the typographical advantage of not requiring, as does v-,
a wider separation of the line in which the sign occurs from the two
lines which comprehend it."
In 1923 the National Committee on Mathematical Requirements1
voiced the following opinion: "Since neither -7- nor :, as signs of di-
vision, plays any part in business life, it seems proper to consider only
the needs of algebra, and to make more use of the fractional form and
(where the meaning is clear) of the symbol /, and to drop the symbol
-7- in writing algebraic expressions. "
244. Notations for geometrical ratio. — William Oughtred intro-
duced in his Clavis mathematicae the dot as the symbol for ratio (§ 181).
He wrote (§ 186) geometrical proportion thus, a.b::c.d. This nota-
tion for ratio arid proportion was widely adopted not only in England,
but also on the European Continent. Nevertheless, a new sign, the
colon (:), made its appearance in England in 1651, only twenty years
after the first publication of Oughtred 's text. This colon is due to the
astronomer Vincent Wing. In 1649 he published in London his
Urania practica, which, however, exhibits no special symbolism for
ratio. But his Harmonicon cocleste (London, 1651) contains many
times Oughtred's notation A.B::C .D, and many times also the new
notation A : B : :C:D, the two notations being used interchangeably.
Later there appeared from his pen, in London, three books in one
volume, Logistica astronomica (1656), Doctrina spherica (1655),
and Doctrina theorica (1655), each of which uses the notation A:B: :
C:D.
A second author who used the colon nearly as early as Wing was a
schoolmaster who hid himself behind the initials "R.B." In his book
entitled An Idea of Arithmetik, at first designed for the use of "the
Free Schoole at Thurlow in Suffolk .... by R.B., Schoolmaster
there" (London, 1655), one finds 1.6: :4.24 and also A:a: :C:c.
W. W. Beman pointed out in Ulntermidiaire des mathematiciens,
Volume IX (1902), page 229, that Oughtred's Latin edition of his
Trigonometria (1657) contains in the explanation of the use of the
tables, near the end, the use of : for ratio. It is highly improbable that
the colon occurring in those tables was inserted by Oughtred himself.
In the Trigonometria proper, the colon does not occur, and Ought-
1 Report of the National Committee on Mathematical Requirements under the
Auspices of the Mathematical Association of America, Inc. (1923), p. 81.
276 A HISTORY OF MATHEMATICAL NOTATIONS
red's regular notation for ratio and proportion A .B: :C .D is followed
throughout. Moreover, in the English edition of Oughtred's trigo-
nometry, printed in the same year (1657), but subsequent to the Latin
edition, the passage of the Latin edition containing the : is recast,
the new notation for ratio is abandoned, and Oughtred's notation is
introduced. The : used to designate ratio (§ 181) in Oughtred's
Opuscula mathematica hactenus inedita (1677) may have been intro-
duced by the editor of the book.
It is worthy of note, also, that in a text entitled Johnsons Arith-
metik; In two Bookes (2d ed.; London, 1633), the colon (:) is used to
designate a fraction. Thus f is written 3:4. If a fraction be con-
sidered as an indicated division, then we have here the use of : for
division at a period fifty-one years before Leibniz first employed it for
that purpose in print. However, dissociated from the idea of a frac-
tion, division is not designated by any symbol in Johnson's text. In
dividing 8976 by 15 he writes the quotient "598 6: 15."
As shown more fully elsewhere (§ 258), the colon won its way as
the regular symbol for geometrical ratio, both in England and the
European Continent.
245. Oughtred's dot and Wing's colon did not prevent experi-
mentation with other characters for geometric ratio, at a later date.
But none of the new characters proposed became serious rivals of the
colon. Richard Balam,1 in 1653, used the colon as a decimal separatrix,
and proceeded to express ratio by turning the colon around so that
the two dots became horizontal; thus "3 . . 1" meant the geometrical
ratio 1 to 3. This designation was used by John Kirkby2 in 1735
for arithmetical ratio; he wrote arithmetical proportion "9.. 6 =
6 . . 3." In the algebra of John Alexander,3 of Bern, geometrical
ratio is expressed by a dot, a. b, and also by a— 6. Thomas York4
in 1687 wrote a geometrical proportion "33600 7 : : 153600 32,"
using no sign at all between the terms of a ratio.
In the minds of some writers, a geometrical ratio was something
more than an indicated division. The operation of division was asso-
ciated with rational numbers. But a ratio may involve incomrnensu-
1 Richard Bulam, Algebra: or The Doctrine of Composing, Inferring, and Re-
solving an Equation (London, 1653), p. 4.
2 John Kirkby, Arithmetical Institutions (London, 1735), p. 28.
3 Synopsis algebraica, opus posthumum lohannis Alexandri, Bernatis-Helvetii.
In usum scholae mathematical apud Hospitium-Christi Londinense (London, 1693),
p. 16, 55. An English translation by Sam. Cobb appeared at London in 1709.
4 Thomas York, Practical Treatise of Arilhmetik (London, 1687), p. 146.
DIVISION AND RATIO 277
rable magnitudes which are expressible by two numbers, one or both
of which are irrational. Hence ratio and division could not be marked
by the same symbol. Oughtred's ratio a.b was not regarded by him
as an indicated division, nor was it a fraction. In 1696 this matter
was taken up by Samuel Jeake1 in the following manner: "And so by
some, to distinguish them [ratios] from Fractions, instead of the in-
tervening Line, two Pricks are set; and so the Ratio Sesquialtera
3
is thus expressed •• ." Jeake writes the geometrical proportion,
z
" 7 ()
Emanuel Swedenborg starts out, in his Daedalus Hyperboreus
(Upsala, 1716), to designate geometric proportion by : : : :, but on
page 126 he introduces —• as a signum analogicum which is really used
as a symbol for the ratio of quantities. On the European Continent
one finds Herigone2 using the letter TT to stand for "proportional"
or ratio; he writes IT where we write : . On the other hand, there are
isolated cases where : was assigned a different usage; the Italian
L. Perini3 employs it as separatrix between the number of feet and of
inches; his "11:4" means 11 feet 4 inches.
246. Discriminating between ratio and division, F. Schmeisser4
in 1817 suggested for geometric ratio the symbol . . , which (as previ-
ously pointed out) had been used by Richard Balam, and which was
employed by Thomas Dil worth5 in London, and in 1799 by Zachariah
Jess,6 of Wilmington, Delaware. Schmeisser comments as follows:
"At one time ratio was indicated by a point, as in a. 6, but as this
signifies multiplication, Leibniz introduced two points, as in a:&,
a designation indicating division and therefore equally inconvenient,
and current only in Germany. For that reason have Monnich, v.
Winterfeld, Krause and other thoughtful mathematicians in more
recent time adopted the more appropriate designation a. .6."
Schmeisser writes (p. 233) the geometric progression: "-J-3..6..12
..24.. 48.. 96 ..... "
1 Samuel Jeake, AOriSTIKITAOrf A, or Arithmetick (London, 1696), p. 410.
2 Peter Herigone, Cursus mcdhematicus, Vol. I (Paris, 1834), p. 8.
3 Lodovico Perini, Gcometria pralica (Venezia, 1750), p. 109.
4 Fricdrich Schmeisser, Lehrbuch der reinen Mathesis, Erster Thcil, "Die
Arithmetik" (Berlin, 1817), Vorrede, p. 58.
5 Thomas Dilworth, The Schoolmaster's Assistant (2d ed.; London, 1784).
(First edition, about 1744.)
6 Zachariah Jess, System of Practical Surveying (Wilmington, 1799), p. 173.
278 A HISTORY OF MATHEMATICAL NOTATIONS
Similarly, A. E. Layng,1 of the Stafford Grammar School in
j^
England, states: 'The Algebraic method of expressing a ratio -~
being a very convenient one, will also be found in the Examples, where
it should be regarded as a symbol for the words the ratio of A to B,
and not as implying the operation of division; it should not be used
for book-work."
247. Division in the algebra of complex numbers. — As, in the alge-
bra of complex numbers, multiplication is in general not commu-
tative, one has two cases in division, one requiring the solution of
a = bx, the other the solution of a = yb. The solution of a = bx is
designated by Peirce2 r--, by Schroder3 -,-, by Study4 and Cartan =-.
The solution of a = yb is designated by Peirce -— ^ and by Schroder
X o
a: 6, by Study and Cartan -T. The X and the . indicate in this nota-
tion the place of the unknown factor. Study and Cartan use also the
notations of Peirce and Schroder.
SIGNS OF PROPORTION
248. Arithmetical and geometrical progression. — The notation -H-
was used by W. Oughtred (§ 181) to indicate that the numbers follow-
ing were in continued geometrical proportion. Thus, -if- 2, 6, 18, 54,
162 are in continued geometric proportion. During the seventeenth
and eighteenth centuries this symbol found extensive application;
beginning with the nineteenth century the need of it gradually
passed away, except among the Spanish-American writers. Among the
many English writers using -ff are John Wallis,5 Richard Sault6,
Edward Cocker,7 John Kersey,8 William Whiston,9 Alexander Mal-
1 A. E. Layng, Euclid's Elements of Geometry (London, 1891), p. 219.
2 B. Peirce, Linear Associative Algebra (1870), p. 17; Amer. Jour, of Math.,
Vol. IV (1881), p. 104.
8 E. Schroder, Formate Elemente der ahsoluten Algebra (Progr. Bade, 1874).
4 E. Study and E. Cartan, Encyclopedic des scien. math., Tom. I, Vol. I (1908),
p. 373.
* Phil. Trans., Vol. V (London, 1670), p. 2203.
6 Richard Sault, A New Treatise of Algebra (London [no date]).
7 Cocker's Artificial Arithmetick, by Edward Cocker, perused and published by
John Hawkes (London, 1684), p. 278.
8 John Kersey, Elements of Algebra (London, 1674), Book IV, p. 177.
9 A. Tacquet's edition of W. Whiston1 s Elemenla Euclidea geometriae (Amster-
dam, 1725), p. 124.
PROPORTION 279
colm,1 Sir Jonas Moore,2 and John Wilson.3 Colin Maclaurin indi-
cates in his Algebra (1748) a geometric progression thus: "-^-liqiq2:
(f:q*:(f\ etc." E. Bezout4 and L. Dcspiau5 write for arithmetical
progression "-v-1.3.5.7.9," and "4f3:6:12" for geometrical pro-
gression.
Symbols for arithmetic progression were less common than for
geometric progression, and they were more varied. Oughtred had no
symbol. Wallis6 denotes an arithmetic progression A, B, C, D ~f,
or by a, 6, c, d, e, /^f. The sign -f-, which we cited as occurring in
Bezout and Despiau, is listed by Saverien7 who writes " -7-1. 2. 3. 4. 5,
etc." But Saverien gives also the six dots :::, which occur in Stone8
and Wilson.9 A still different designation, •—-, for arithmetical pro-
gression is due to Kirkby10 and Emerson,11 another -Hr- to Clark,12 again
another -fr is found in Blassiere.13 Among French writers using -~ for
arithmetic progression and -H- for geometric progression are Lamy,14
De Belidor,15 Suzanne,16 and Fournier;17 among Spanish-American
1 Alexander Malcolm, A New System of Arithmetick (London, 1730), p. 115.
2 Sir Jonas Moore, Arithmetick in Four Books (3d ed.; London, 1688), begin-
ning of the Book IV.
3 John Wilson, Trigonometry (Edinburgh, 1714), p. 24.
4E. Bezout, Cours de mathtmatiques, Tome I (2. 6d.; Paris, 1797), "Arith-
m6tiquc," p. 130, 165.
6 Select Amusements in Philosophy of Mathematics .... translated from the
French of M. L. Dcspiau, Formerly Professor of Mathematics and Philosophy
at Paris Recommended .... by Dr. Hutton (London, 1801), p. 19, 37, 43.
6 John Wallis, Operum mathematicorvm Pars Prima (Oxford, 1657), p. 230, 236.
7 A. Saverien, Dictionnaire universel de mathematique et de physique (Paris,
1753), art. "Caractere."
8 E. Stone, New Mathematical Dictionary (London, 1726), art. "Characters."
9 John Wilson, Trigonometry (Edinburgh, 1714).
10 John Kirkby, Arithmetical Institutions containing a compleat System of
Arithmetic (London, 1735), p. 36.
u W. Ernerson, Doctrine of Proportion (1763), p. 27.
w Gilbert Clark, Oughtredus explicatus (London, 1682), p. 114.
13 J. J. Blassiere, Institution du calcul numerique et litteral (a La Haye, 1770),
end of Part II.
14 Bernard Lamy, Siemens des mathematiques (3d ed. ; Amsterdam, 1692),
p. 156.
18 B. F. de Belidor, Nouveau Cours de mathematique (Paris, 1725), p. 71, 139.
16 H. Suzanne, De la Maniere d' eludier1, es Mathematiques (2. &L; Paris, 1810),
p. 208.
» C. F. Fournier, Elements d' Arithmttique et d'Algebre, Vol. II (Nantes, 1822).
280 A HISTORY OF MATHEMATICAL NOTATIONS
writers using these two symbols are Senillosa,1 Izquierdo,2 Lidvano,3
and Porfirio da Motta Pegado.4 In German publications ~ for arith-
metical progression and ^f for geometric progression occur less fre-
quently than among the French. In the 1710 publication in the Mis-
cellanea Berolinensia5 -ff is mentioned in a discourse on symbols
(§ 198). The -=f was used in 1716 by Emanuel Swedenborg.6
Emerson7 designated harmonic progression by the symbol -^v and
harmonic proportion by .V. .
249. Arithmetical proportion finds crude symbolic representation
in the Arithmetic of Boethius as printed at Augsburg in 1488 (see
Figure 103). Being, in importance, subordinate to geometrical pro-
portion, the need of a symbolism was less apparent. But in the seven-
teenth century definite notations came into vogue. William Oughtred
appears to have designed a symbolism. Oughtred's language (Clavis
[1652], p. 21) is "Ut 7.4:12.9 vel 7.7-3:12.12-3. Arithmetics
proportionales sunt." As later in his work he does not use arithmetical
proportion in symbolic analysis, it is not easy to decide whether the
symbols just quoted were intended by Oughtred as part of his alge-
braic symbolism or merely as punctuation marks in ordinary writing.
John Newton8 says: "As 8,5:6,3. Here 8 exceeds 5, as much as 6
exceeds 3."
Wallis9 says: "Et pariter 5,3; 11,9; 17,15; 19,17. sunt in cadcm
progressione arithmetica." In P. Chelucci's10 Inslitutiones analyticae,
arithmetical proportion is indicated thus: 6.8'. '10. 12. Oughtred's
notation is followed in the article "Caractere" of the Encydop6die
Felipe Senillosa, Tratado elemental de Arismdica (Ncuva ed.; Buenos Aires,
1844), p. 46.
2 Gabriel Izquierdo, Tratado de Aritmetica (Santiago [Chile], 1859), p. 167.
3 Indalecio Lidvano, Tratado de Aritmetica (2. e\L; Bogota, 1872), p. 147.
4 Luiz Porfirio da Motta Pegado, Tratade elementar de arithmetica (2. e*d. ;
Lisboa, 1875), p. 253.
6 Miscellanea Berolinensia (Berolini, 1710), p. 159.
6 Emanuel Swedberg, Daedalus hyperboreus (Upsala, 1716), p. 126. Facsimile
reproduction in Kungliga Vetenskaps Societetens i Upsala Tvahundraarsminne
(Upsala, 1910).
7 W. Emerson, Doctrine of Proportion (London, 1763), p. 2.
8 John Newton, Institutio mathematica or mathematical Institution (London,
1654), p. 125.
9 John Wallis, op. cit. (Oxford, 1657), p. 229.
10Paolino Chelucci, Institutiones analyticae (editio post tertiam Romanam
prima in Germania; Vienna, 1761), p. 3. See also the first edition (Rome, 1738),
p. 1-15.
PROPORTION 281
mtthodique (Mathtmatiques) (Paris: Li£ge, 1784). Lamy1 says:
"Proportion arithm&ique, 5,7 Y 10,12.c'est & dire qu'il y a m£me
difference entre 5 et 7, qu'entre 10 et 12."
In Arnauld's geometry2 the same symbols are used for arithmeti-
cal progression as for geometrical progression, as in 7.3:: 13.9 and
6.2::12.4.
Samuel Jeake (1696)3 speaks of " • Three Pricks or Points, some-
times in disjunct proportion for the words is as."
A notation for arithmetical proportion, noticed in two English
seventeenth-century texts, consists of five dots, thus :•:; Richard
Balam4 speaks of "arithmetical disjunct proportionals" and writes
"2.4 :-:3.5"; Sir Jonas Moore5 uses :•: and speaks of "disjunct pro-
portionals." Balam adds, "They may also be noted thus, 2... 4 =
3... 5." Similarly, John Kirkby6 designated arithmetrical propor-
tion in this manner, 9.. 6 = 6.. 3, the symbolism for arithmetical
ratio being 8. .2. L'Abb6 Deidier (1739)7 adopts 20. 2. \78.60. Be-
fore that Weigcl8 wrote "(o) 3| V 4.7" and "(o) 2.| V 3.5." Wolff
(1710),9 Panchaud,10 Savcrien,11 L'AbbS Foucher,12 Emerson,13 place
1 B. Lamy, Elemens dea mathemotiques (3. 6d.; Amsterdam, 1692), p. 155.
2Antoine Arnauld, Nouveaux elemem de geometric (Paris, 1667); also in the
edition issued at The Hague in 1690.
» Samuel Jeake, AOriSTIKIIAOriA or Arithmetick (London, 1696; Preface,
1674), p. 10-12.
4 Richard Balam, Algebra: or the Doctrine of Composing, Inferring, and Re-
solving an Equation (London, 1653), p. 5.
5 Sir Jonas Moore, Moore's Arithmetick: In Four Books (3d ed.; London, 1688),
the beginning of Book IV.
8 Rev. Mr. John Kirkby, Arithmetical Institutions containing a compleat Sys-
tem of Arithmetic (London, 1735), p. 27, 28.
7 L'Abbc" Deidier, L'Arithm&iques des geometrest ou nouveau elemens de malhe-
matiques (Paris, 1739), p. 219.
8 Erhardi Weigelii Specimina novarum inveniionum (Jenae, 1693), p. 9.
9 Chr. v. Wolff, Anfangsgrunde oiler math. Wissenschaften (1710), Vol. I, p. 65.
See J. Tropfke, op. cit., Vol. Ill (2d ed., 1922), p. 12.
10 Benjamin Panchaud, Entreliens ou lecons mathematiques, Premier Parti
(Lausanne et Geneve, 1743), p. vii.
11 A. Saverien, Dictionnaire universel (Paris, 1753), art. "Proportion arith-
metique."
11 L'Abb6 Foucher, Geometrie metaphysique ou essai df analyse (Paris, 1758),
p. 257
» W. Emerson, The Doctrine of Proportion (London, 1763), p. 27.
282 A HISTORY OF MATHEMATICAL NOTATIONS
the three dots as did Chelucci and Deidier, viz., a.b'.'c.d. Cosalli1
writes the arithmetical proportion a : b V c:d. Later Wolff2 wrote a— 6
= c— d.
Blassiere3 prefers 2:7-rrlO:15. Juan Gerard4 transfers Oughtred's
signs for geometrical proportion to arithmetical proportion and
writes accordingly, 9.7:: 5.3. In French, Spanish, and Latin- Ameri-
can texts Oughtred's notation, 8.6:5.3, for arithmetical proportion
has persisted. Thus one finds it in Benito Bails,6 in a French text for
the military,6 in Fournier,7 in Gabriel Izquierdo,8 in Indalecio Lievano.9
250. Geometrical proportion. — A presentation of geometrical pro-
portion that is not essentially rhetorical is found in the Hindu Bakh-
shilli arithmetic, where the proportion 10: JCV = 4: }§f is written in
the form10
10
1
163
60
4
1
pha 163
150
It was shown previously (§ 124) that the Arab al-Qalasddi (fifteenth
century) expresses the proportion 7.12 = 84:144 in this manner:
144 /. 84 /. 12 /. 7. Regiomontanus in a letter writes our modern
a : b : c in the form a.b.c, the dots being simply signs of separation.11 In
the edition of the Arithmetica of Boethius, published at Augsburg in
1488, a crude representation of geometrical and arithmetical propor-
1 Scritli inedili del P. D. Pielro Cossali .... pubblicati da B. Boncompagni
(Rome, 1857), p. 75.
2 Chr. v. Wolff., op. cit. (1750), Vol. I, p. 73.
3 J. J. Blassiere, Institution du calcul numerique et lateral (a La Haye 1770),
the end of Part II.
4 Juan Gerard, Tratado complete de aritmetica (Madrid, 1798), p. 69.
6 Benito Bails, Principios de matematica de la real academia de San Fernando
(2. ed.), Vol. I (Madrid, 1788), p. 135.
6 Cours de mathSmaliques, d V usage des ecoles imperiales militaires ... r6dige"
par ordre de M. le Ge'ne'ral de Division Bellavene ... (Paris, 1809), p. 52. Dedica-
tion signed by "Allaize, Billy, Puissant, Boudrot, Professeurs de mathdmatiques a
TEcole de Saint-Cyr."
7 C. F. Fournier, Elements d' arithmetique et d'algebre. Tome II (Nantes, 1842),
p. 87.
8 Gabriel Izquierdo, op. cit. (Santiago [Chile], 1859), p. 155.
9 Indalecio Li6vano, Tratado aritmetica (2d ed.; Bogota, 1872), p. 147.
10 G. R. Kaye, The Bakhshall Manuscript, Parts I and II (Calcutta, 1927),
p. 119.
" M. Curtze, Abhandlungen z. Geschichte d. Mathematik, Vol. XII (1902),
p. 253.
PROPORTION
283
tion is given, as shown in Figure 103. The upper proportion on the
left is geometrical, the lower one on the left is arithmetical. In the
latter, the figure 8 plays no part; the 6, 9, and 12 are in arithmetical
proportion. The two exhibitions on the right relate to harmonical
and musical proportion.
Proportion as found in the earliest printed arithmetic (in Treviso,
<£ttpttfciitf fen iraoitfct.
0 o<wi «nu mtoiu nunp
&.a*»fcer«mirwc.
ttfoei.ii.faciA.'i-Qti
riu«Aiper«f.Oiiatuot Stf^idUcrfj^omone*.
MiarfiM a fruofccnano v-^ — > V^^X.
S£SgF»£S f f\ \
ii«l><fimnipUee«. »44- H < I > I » i «» I
rtreirtrwc*inttiplKft: i — V / F"^
aei&.r».qtionum«ro I S^ J
rtrtCf mftlOtlf maiMifiB ftniitqr. PWT« per £rbwbtir«tboUvi
i _4 icrturimicxtmiaitnirtnatmkaim/
ougtiflcerceUct nominatUTunu*.
.
dttwficM confonannw.
If Ktfqwirerti i,ppouion£ re^ui . t (I
•ntaiHa:ertcron ofoiiantul. Sejtfo
^.9.wl.8.ot>. a cM>arari rtt>t>ut
a b c d c f
ftinpbonli.|>uot)ee»mf o^ ftnav
*44
Jimctf enrenu«3K» t par
tn^iamulnplicate.
f o t.f.irfi ?ir<i ft nxNj conRtterart
jpocbou... - - - -
•u mugii.fl in mufi<o motm
tooottcxaf.qii "
coriifortojurmnfurdc
.
tooot vocof .qiit 0(0} mull/
d Jmoni* eft.
ttftttftff.
Untxnort cftq6 MaWeront bi«
Mnic sfononriarii io.to* biffrrfti«
%:lkui inter fcfcuiicrciit fcfqiMl/
nf foU < epocboiw Mf
FIG. 103. — From the Arithmetica of Boethius, as printed in 1488, the last two
pages. (Taken from D. K. Smith's Rara arithmetica [Boston, 1898], p. 28.)
1487) is shown in Figure 39. Stifel, in his edition of Rudolff s Coss
(1553), uses vertical lines of separation, as in
"100 | i- z I 100 z | Facit J zz ."
Tartaglia1 indicates a proportion thus:
<(Se X 3// val 0 4 // che valeranno £ 28."
Chr. Clavius2 writes:
"9 . 126 . 5 . ? fiunt 70 ."
1 N. Tartaglia, La prima parte del General Tratato di Nvmeri, etc. (Venice
1556), fol. 1291?.
2 Chr. Clavius, Epitome arithmeticae practicae (Rome, 1583), p. 137.
284 A HISTORY OF MATHEMATICAL NOTATIONS
This notation is found as late as 1699 in Corachan's arithmetic1 in
such statements as
"A . B . C . D .
5 . 7 . 15 . 21 . "
Schwenter2 marks the geometric proportion 68 51 85, then
finds the product of the means 51X85 = 4335 and divides this by 68.
In a work of Galileo,3 in 1635, one finds:
"Regula aurea
58 95996 . 21600 .
21600
57597600
95996
191992
58
357
20735
3339
42"
13600
In other places in Galileo's book the three terms in the proportion
are not separated by horizontal lines, but by dots or simply by spac-
ing. Johan Stampioen,4 in 1639, indicates our a:6 = 6:c by the sym-
bolism:
"a,, b gel : b „ c .»
Further illustrations are given in § 221.
These examples show that some mode of presenting to the eye
the numbers involved in a geometric proportion, or in the applica-
tion of the rule of three, had made itself felt soon after books on mathe-
matics came to be manufactured. Sometimes the exposition was rhe-
torical, short words being available for the writing of proportion. As
late as 1601 Philip Lansberg5 wrote "ut 5 ad 10; ita 10 ad 20," meaning
1 Ivan Bavtista Corachan, Arilhmetica demonslrada (Valencia, 1699), p. 199.
2 Daniel Schwentcr, Geomclriae practicae novae et auctac tractatus (Numbers,
1623), p. 89.
3 Syslema Cosmicvm, aucthore Galilaeo Galilaei Ex Italica lingua f aline
conversum (Florence, 1635), p. 294.
4 Johan Stampioen, Algebra ofte nieuwe Slel-Regel (The Hague, 1639), p. 343.
6 Philip Lansberg, Triangulorum geomelriae libri quaiuor (Middelburg [Zee-
land], 1663), p. 5.
PROPORTION 285
5:10 = 10:20. Even later the Italian Cardinal Michelangelo Ricci1
wrote "esto AC ad CB, ut 9 ad 6." If the fourth term was not given,
but was to be computed from the first three, the place for the fourth
term was frequently left vacant, or it was designated by a question
mark.
251. Oughtred' s notation. — As the symbolism of algebra was being
developed and the science came to be used more extensively, the need
for more precise symbolism became apparent. It has been shown
(§ 181) that the earliest noted symbolism was introduced by Ought-
red. In his Clavis mathematicae (London, 1631) he introduced the
notation 5 . 10 : : 6 . 12 which he retained in the later editions of this
text, as well as in his Circles of Proportion (1632, 1633, 1660), and in
his Trigonometria (1657).
As previously stated (§ 169) the suggestion for this symbolism may
have come to Oughtred from the reading of John Dee's Introduction
to Billingley's Euclid (1570). Probably no mathematical symbol has
been in such great demand in mathematics as the dot. It could be used,
conveniently, in a dozen or more different meanings. But the avoid-
ance of confusion necessitates the restriction of its use. Where then
shall it be used, and where must other symbols be chosen? Oughtred
used the dot to designate ratio. That made it impossible for him to fol-
low John Napier in using the dot as the scparatrix in decimal fractions.
Oughtred could not employ two dots (:) for ratio, because the two
dots were already pre-empted by him for the designation of aggre-
gation, :A-\-B: signifying (A-{-B). Oughtred reserved the dot for
the writing of ratio, and used four dots to separate the two equal
ratios. The four dots were an unfortunate selection. The sign of
equality ( = ) would have been far superior. But Oughtred adhered to
his notation. Editions of his books containing it appeared repeatedly
in the seventeenth century. Few symbols have met with more
prompt adoption than those of Oughtred for proportion. Evidently
the time was ripe for the introduction of a definite unambigu-
ous symbolism. To be sure the adoption was not immediate. Nine-
teen years elapsed before another author used the notation A.B::
C .D. In 1650 John Kersey brought out in London an edition of
Edmund Wingate's Arithmetique made easie, in which this notation is
used. After this date, the publications employing it became frequent,
some of them being the productions of pupils of Oughtred. We have
1 Michaelis Angeli Rictii exercitatio geometrica de maximis et minimis (London,
1068), p. 3.
286 A HISTORY OF MATHEMATICAL NOTATIONS
seen it in Vincent Wing,1 Seth Ward,2 John Wallis,3 in "R.B.," a
schoolmaster in Suffolk,4 Samuel Foster,5 Sir Jonas Moore,6 and Isaac
Barrow.7 John Wallis8 sometimes uses a peculiar combination of
processes, involving the simplification of terms, during the very act
of writing proportion, as in ' '%A = 4A . %A = 3 A : : %A = 2 A • f A : : 8 . 6 : :
4.3." Here the dot signifies ratio.
The use of the dot, as introduced by Oughtred, did not become
universal even in England. As early as 1651 the astronomer, Vincent
Wing (§ 244), in his Harmonicon Coeleste (London), introduced the
colon ( : ) as the symbol for ratio. This book uses, in fact, both nota-
tions for ratio. Many times one finds A.B'.'.C.D and many times
A:B::C:D. It may be that the typesetter used whichever notation
happened at the moment to strike his fancy. Later, Wing published
three books (§ 244) in which the colon (:) is used regularly in writing
ratios. In 1655 another writer, "R.B.," whom we have cited as using
the symbols A.B::C.D, employed in the same publication also
A:B::C:D. The colon was adopted in 1661 by Thomas Streete.9
That Oughtred himself at any time voluntarily used the colon as
the sign for ratio does not appear. In the editions of his Clavis of
1648 and 1694, the use of : to signify ratio has been found to occur
only once in each copy (§ 186) ; hence one is inclined to look upon this
notation in these copies as printer's errors.
252. Struggle in England between Oughtred's and Wing's notations,
before 1 700. — During the second half of the seventeenth century there
was in England competition between (.) and (:) as the symbols for
the designation of the ratio (§§ 181, 251). At that time the dot main-
tained its ascendancy. Not only was it used by the two most influ-
1 Vincent Wing, Harmonicon coeleste (London, 1651), p. 5.
2 Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenla in-
quisitio brevis (Oxford, 1653), p. 7.
3 John Wallis, Elenchus geometriae Hobbianac (Oxford, 1655), p. 48; Operum ma-
thematicorum pars alter a (Oxford, 1656), the part on Arithmetica infinitorum, p. 181.
4 An Idea of ArUhmetick, at first designed for the use of the Free Schoole at
Thurlow in Suffolk By R. B., Schoolmaster there (London, 1655), p. 6.
6 Miscellanies: or mathematical Lucrubations of Mr. Samuel Foster .... by
John Twyden (London, 1659), p. 1.
6 Jonas Moore, Arilhmelick in two Books (London, 1660), p. 89; Moore's
Arithmetique in Four Books (3d ed.; London, 1688), Book IV, p. 401.
7 Isaac Barrow's edition of Euclid's Data (Cambridge, 1657), p. 2.
8 John Wallis, Adversus Marci Meibomii de Proportionibus Dialogum (Oxford,
1657), "Dialogum," p. 54.
9 Thomas Streete, Astronomia Carolina (1661). See J. Tropfke, Geschichte der
Elcmentar-Mathematik, 3. Bd., 2. Aufl. (Berlin und Leipzig, 1922), p. 12.
PROPORTION 287
ential English mathematicians before Newton, namely, John Wallis
and Isaac Barrow, but also by David Gregory,1 John Craig,2 N.
Mercator,3 and Thomas Brancker.4 I. Newton, in his letter to Olden-
burg of October 24, 1676,5 used the notation . :: . , but in Newton's
De analyxi per aequationes tcrminorum infinitas, the colon is employed
to designate ratio, also in his Quadratures curvarum.
Among seventeenth-century English writers using the colon to mark
ratio are James Gregory,6 John Collins,7 Christopher Wren,8 William
Leybourn,9 William Sanders,10 John Hawkins,11 Joseph Raphson,12
E. Wells,13 and John Ward.14
253. Struggle in England between Oughtred's and Wing's notations
during 1700-1750. — In the early part of the eighteenth century, the
dot still held its place in many English books, but the colon gained in
ascendancy, and in the latter part of the century won out. The single
dot was used in John Alexander's Algebra (in which proportion is
written in the form a.biic.X and also in the form a~b:c~X)u and,
in John Colson's translation of Agnesi (before 1760). 16 It was used
1 David Gregory in Phil. Trans., Vol. XIX (1G95-97), p. 645.
2 John Craig, Methodus fiyurarum lineis rectis et curvis (London, 1G85). Also
his Traclatus mathematicus (London, 1G93), but in 1718 he often used : : : .in
his De Calculo Fluentium Libri Duo, brought out in London.
3 N. Mercator, Logarithmotechnia (London, 1668), p. 29.
4 Th. Brancker, Introduction to Algebra (trans, of Rhonius; London, 1668),
p. 37.
5 John Collins, Commercium epistolicum (London, 1712), p. 182.
6 James Gregory, Vera circuli et hyperbolae quadratura (Patavia, 1668), p. 33.
7 J. Collins, Mariners Plain Scale New Plain' d (London, 1659).
8 Phil. Trans., Vol. Ill (London), p. 868.
9W. Leybourn, The Line of Proportion (London, 1673), p. 14.
10 William Sanders, Elementa geometriae (Glasgow, 1686), p. 3.
11 Cocker's Decimal Arithmetick .... perused by John Hawkins (London,
1695) (Preface dated 1684), p. 41.
12 J. Haphson, Analysis aequationum universalis (London, 1697), p. 26.
13 E. Wells, Elementa arithmeticae numerosae et speciosae (Oxford, 1698), p. 107.
14 John Ward, A Compendium of Algebra (2d ed.; London, 1698), p. 62.
15 A Synopsis of Algebra. Being the Posthumous Work of John Alexander, of
Bern in Swisserland. To which is added an Appendix by Humfrey Ditton
Done from the Latin by Sam. Cobb, M.A. (London, 1709), p. 16. The Latin edi-
tion appeared at London in 1693.
16 Maria Gaetana Agnesi, Analytical Institutions, translated into English by the
late Rev. John Colson Now first printed .... under the inspection of Rev.
John Hellins (London, 1801).
288 A HISTORY OF MATHEMATICAL NOTATIONS
by John Wilson1 and by the editors of Newton's Universal arithmetick?
In John Harris' Lexicon technicum (1704) the dot is used in some
articles, the colon in others, but in Harris' translation3 of G. Pardies*
geometry the dot only is used. George Shelley4 and Hatton5 used the
dot.
254. Sporadic notations. — Before the English notations . : : . and
: :: : were introduced on the European Continent, a symbolism con-
sisting of vertical lines, a modification of Tartaglia's mode of writing,
was used by a few continental writers. It never attained popularity,
yet maintained itself for about a century. Ren6 Descartes (1619-21)°
appears to have been the first to introduce such a notation a|fe||c|d.
In a letter7 of 1638 he replaces the middle double stroke by a single
one. Slusius8 uses single vertical lines in designating four numbers in
geometrical proportion, p \ a \ e \ d — a. With Slusius, two vertical strokes
1 1 signify equality. Jaques de Billy9 marks five quantities in continued
proportion, thus 3 — #5|#5 — 1| 2|#5+1 |3+#5, where & means
"square root." In reviewing publications of Huygens and others, the
original notation of Descartes is used in the Journal des Sgavans
(Amsterdam)10 for the years 1701, 1713, 1716. Likewise, Picard,11 De la
Hire,12 Abraham de Graaf ,13 and Parent14 use the notation a \ b \ \ xx \ ab.
I John Wilson, Trigonometry (Edinburgh, 1714), p. 24.
2 1. Newton, Arithmetica universalis (ed. W. Whiston; Cambridge, 1707),
p. 9; Universal Arithmetick, by Sir Isaac Newton, translated by Mr. llalphson
.... revised .... by Mr. Cunn (London, 1769), p. 17.
3 Plain Elements of Geometry and Plain Trigonometry (London, 1701), p. 63.
4 G. Shelley, Wingate's Arithmetick (London, 1704), p. 343.
6 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 93.
6 (Euvres des Descartes (e*d. Adam et Tannery), Vol. X, p. 240.
7 Op. dt., Vol. II, p. 171.
8 Renati Francisci Slusii mesolabum seu duae mediae proportionates, etc.
(1668), p. 74. See also Slusius' reply to Huygens in Philosophical Transactions
(London), Vols. III-IV (1668-69), p. 6123.
9 Jaques de Billy, Nova geometriae clavis (Paris, 1643), p. 317.
10 Journal des S^avans (Amsterdam, ann6e 1701), p. 376; ibid. (annc*e 1713),
p. 140, 387; ibid. (ann6e 1716), p. 537.
II J. Picard in Memoires de I'Acadtmie r. des sciences (depuis 1666 jusqu'a
1699), Tome VI (Paris, 1730), p. 573.
12 De la Hire, Nouveaux elemens des sections coniques (Paris, 1701), p. 184.
J. Tropfke refers to the edition of 1679, p. 184.
18 Abraham de Graaf, De vervulling van der geomelria en algebra (Amsterdam,
1708), p. 97.
14 A. Parent, Essais et recherches de mathemalique et de physique (Paris, 1713),
p. 224.
PROPORTION 289
It is mentioned in the article "Caractere" in Diderot's Encyclopedic
(1754) . La Hire writes also "aa \ \ xx \ \ ab" for a2 : re2 = a:2 : ab.
On a subject of such universal application in commercial as well as
scientific publications as that of ratio and proportion, one may expect
to encounter occasional sporadic attempts to alter the symbolism.
Thus Herigone1 writes "hg TT ga 2|2 hb IT bd, signifi. HG est ad GA, vt
HB ad BD," or, in modern notation, hg:ga — hb:bd; here 2 1 2 signifies
equality, TT signifies ratio. Again Peter Mengol,2 of Bologna, writes
"a;r:a2;ar" for a:r — a?:ar. The London edition of the algebra of the
Swiss J. Alexander3 gives the signs . :: . but uses more often designa-
tions like b~a:
— . Ade Mercastel,4 of Rouen, writes 2,, 3 ;;8,,12. A
close approach to the marginal symbolism of John Dee is that of the
Spaniard Zaragoza5 4.3:12.9. More profuse in the use of dots is
TT
J. Kresa6 who writes x. . .r::r. . .-, also AE. .EF::AD. .DG. The
latter form is adopted by the Spaniard Cassany7 who writes 128. . 119
:: 3876; it is found in two American texts,8 of 1797.
In greater conformity with pre-Oughtredian notations is van
Schooten's notation9 of 1657 when he simply separates the three given
numbers by two horizontal dashes and leaves the place for the
unknown number blank. Using Stevin's designation for decimal
Ib. flor. Ib.
fractions, he writes "65 95,753® -1." Abraham de Graaf10 is
1 Pierre Herigone, Cvrsvs mathematici (Paris, 1644), Vol. VI, "Explicatio
notarum." The first edition appeared in 1642.
2 Pietro Mengoli, Geometriae speciosae elementa (Bologna, 1659), p. 8.
3 Synopsis algcbraica, Opus posthumum Johannis Alcxandri, Bcrnatis-Hel-
vctii (London, 1693), p. 135.
4 Jean Baptiate Adrien de Mercastel, Arithmetique cUmontree (Rouen, 1733),
p. 99.
6 Joseph Zaragoza, Arithmetica universal (Valencia, 1669), p. 48.
6 Jacob Kresa, Analysis speciosa trigonometriae sphericae (Prague, 1720),
p. 120, 121.
7 Francisco Cassany, Arithmetica Deseada (Madrid, 1763), p. 102.
8 American Tutor's Assistant. By sundry teachers in and near Philadelphia
(3d ed.; Philadelphia, 1797), p. 57, 58, 62, 91-186. In the "explanation of char-
acters," : : : : is given. The second text is Chauncey Lee's American Accomptant
(Lansingburgh, 1797), where one finds (p. 63) 3. .5: : 6. .10.
9 Francis a Schooten, Leydensis, Exercitationum mathematicarum liber primus
(Leyden, 1657), p. 19.
10 Abraham de Graaf, De Geheele mathesia ofwiskonst (Amsterdam, 1694), p. 16.
290 A HISTORY OF MATHEMATICAL NOTATIONS
partial to the form 2 — 4 = 6—12. Thomas York1 uses three dashes
125—429—10—?, but later in his book writes "33600 7 :: 153600 32,"
the ratio being here indicated by a blank space. To distinguish
ratios from fractions, Samuel Jeake2 states that by some authors
"instead of the intervening Line, two Pricks are set; and so the Ratio
3 11
sesquialtera is thus expressed •• ." Accordingly, Jeake writes " • • • ••
::9.7."
In practical works on computation with logarithms, and in some
arithmetics a rhetorical and vertical arrangement of the terms of a
proportion is found. Mark Forster3 writes:
"As Sine of 40 deg. 9 , 8080675
To 1286 3,1092401
So is Radius 10,0000000
To the greatest Random 2000 3 , 301 1726
Or, For Random at 36 deg."
As late as 1789 Benjamin Workman4 writes " ^L^lnsj '"
255. Oughtred's notation on the European Continent. — On the Euro-
pean Continent the dot as a symbol of geometrical ratio, and the four
dots of proportion, . :: ., were, of course, introduced later than in
England. They were used by Dulaurens,5 Prestet,6 Varignon,7 Pardies,8
De THospital,9 Jakob Bernoulli,10 Johann Bernoulli,11 Carr6,12 Her-
1 Thomas York, Practical Treatise of Arithmetick (London, 1687), p. 132, 146.
2 Samuel Jeake, AOriSTIKHAOFlA, or Arithmetick (London, 1696 [Preface,
1674]), p. 411.
3 Mark Forster, Arithmetical Trigonometry (London, 1690), p. 212.
4 Benjamin Workman, American Accountant (Philadelphia, 1789), p. 62.
6 Francisci Dulaurens, Specimina mathematica (Paris, 1667), p. 1.
6 Jean Prestet, Siemens des mathematiques (Preface signed "J.P.") (Paris,
1675), p. 240. Also Nouveaux elemens des mathematiques, Vol. I (Paris, 1689),
p. 355.
7 P. Varignon in Journal des S^avans, ann6e 1687 (Amsterdam, 1688), p. 644.
Also Varignon, Eclair cissemens sur V analyse des infiniment petits (Paris, 1725),
p. 16.
8 (Euvres du R. P. Ignace-Gaston Pardies (Lyon, 1695), p. 121.
9 De 1'Hospital, Analyse des infiniment petits (Paris, 1696), p. 11.
10 Jakob Bernoulli in Acta eruditorum (1687), p. 619 and many other places.
11 Johann Bernoulli in Histoire de V academic r. des sciences, ann6e 1732 (Paris,
1735), p. 237.
12 L. Carre", Methode pour la Mesure des Surfaces (Paris, 1700), p. 5.
PROPORTION 291
maim,1 and Rollc;2 also by De Reaumur,3 Saurin,4 Parent,5 Nicole,6
Pitot,7 Poleni,8 De Mairan,9 and Maupertuis.10 By the middle of the
eighteenth century, Oughtred's notation A.B::C .D had disappeared
from the volumes of the Paris Academy of Sciences, but we still find
it in textbooks of Belidor,11 Guido Grandi,12 Diderot,13 Gallimard,14
De la Chapelle,15 Fortunato,16 L'Abbe Foucher,17 and of Abb£ Girault
de Koudou.18 This notation is rarely found in the writings of German
authors. Erhard Weigel19 used it in a philosophical work of 1693.
Christian Wolf20 used the notation "DC .AD:: EC .ME" in 1707, and
in 1710 "3 . 12 : : 5 . 20" and also "3 : 12 = 5 : 20." Beguelin21 used the dot
for ratio in 1773. From our data it is evident that A.B::C .D began
1 J. Hermann in Ada eruditorum (1702), p. 502.
2 M. Rollc in Journal des S$avans, ann6e 1702 (Amsterdam, 1703), p. 399.
3 R. A. F. de Reaumur, Histoire de V academic r. des sciences, annc*e 1708
(Paris, 1730), "Me" moires," p. 209, but on p. 199 he used also the notation : : : :.
4 J. Saurin, op. ciL, ann6e 1708, "M6moires," p. 26.
5 Antoine Parent, op. cit., annee 1708, "Mcmoircs," p. 118.
6F. Nicole, op. cit., amide 1715 (Paris, 1741), p. 50.
7 II. Pitot, op. tit., amide 1724 (Paris, 1726), "M6moires," p. 109.
8 Joannis Poleni, Epislolarvm mathcmaticarvm Fascicvlvs (Patavii, 1729).
9 J. J. de Mairan, Histoire de I'academie r. des sciences, anndc 1740 (Paris, 1742),
p. 7.
10 P. L. Maupertuis, op. tit., ann6e 1731 (Paris, 1733), "M&noircs," p. 465.
11 B. F. de Belidor, Nouveau Cours de mathematique (Paris, 1725), p. 481.
12 Guido Grandi, Elementi geometrici piani e solide de Euclide (Florence, 1740).
13 Deny s Diderot, Memoir -es sur differens sujets de Mathematiques (Paris, 1748),
p. 16.
14 J. E. Gallimard, Geometrie elementaire d 'Euclide (nouvelle 6d. ; Paris, 1749),
p. 37.
16 De la Chapelle, Traite des sections coniqucs (Paris, 1750), p. 150.
16 F. Fortunato, Elementa matheseos (Brescia, 1750), p. 35.
17 L'Abb6 Foucher, Geometric metaphy$ique ou Essai d' analyse (Paris, 1758),
p. 257.
18 L'Abb6 Girault de Koudou, Lemons analytiques du calcul des fluxions et des
fluentes (Paris, 1767), p. 35.
19 Erhardi Weigelii Philosophia mathematica (Jenae, 1693), "Specimina no-
varum inveiitionum," p. 6, 181.
20 C. Wolf in Ada eruditorum (1707), p. 313; Wolf, Anfangsgriinde aller mathe-
matischen Wissemchaften (1710), Band I, p. 65, but later Wolf adopted the nota-
tion of I^eibniz, viz., A:B — C:D. See J. Tropfke, Geschichte der Elementar-Mathe-
matik, Vol. Ill (2d ed.; Berlin und Leipzig, 1922), p. 13, 14.
21 Nicolas dc Beguelin in Nouvcaux mtmoires de I'academie r. des sciences et
belles-lettres, annexe 1773 (Berlin, 1775), p. 211.
292 A HISTORY OF MATHEMATICAL NOTATIONS
to be used in the Continent later than in England, and it was also
later to disappear on the Continent.
256. An unusual departure in the notation for geometric propor-
tion which involved an excellent idea was suggested by a Dutch
author, Johan Stampioen,1 as early as the year 1639. This was only
eight years after Oughtrcd had proposed his . :: . Stampioen uses
the designation A, ,# = C, ,D. We have noticed, nearly a century
later, the use of two commas to represent ratio, in a French writer,
Mercastel. But the striking feature with Stampioen is the use of
Recorde's sign of equality in writing proportion. Stampioen antici-
pates Leibniz over half a century in using = to express the equality of
two ratios. He is also the earliest writer that we have seen on the
European Continent to adopt Recorde's symbol in writing ordinary
equations. He was the earliest writer after Descartes to use the ex-
ponential form a3. But his use of = did not find early followers. He
was an opponent of Descartes whose influence in Holland at that
time was great. The employment of = in writing proportion appears
again with James Gregory2 in 1668, but he found no followers in this
practice in Great Britain.
257. Slight modifications of Oughtred's notation. — A slight modifica-
tion of Oughtred's notation, in which commas took the place of the
dots in designating geometrical ratios, thus A , B : : C , D, is occasionally
encountered both in England and on the Continent. Thus Sturm3
4bb 4b
writes "36 , 26 : : 26, -^g- sive ~ /' Lamy4 "3,6:: 4, 8," as did also
Ozanam,5 De Moivre,6 David Gregory,7 L'Abb6 Deidier,8 Belidor,9
who also uses the regular Oughtredian signs, Maria G. Agnesi,10
1 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel ('& Graven-Have,
1639).
2 James Gregory, Geometriae Pars Vniversalis (Padua, 1668), p. 101.
3 Christopher Sturm in Ada erudilorum (Leipzig, 1685), p. 260.
4 R. P. Bernard Lamy, Elcmens dez mathematiques, troisieme edition revue
ct augmented sur PimprismS a Paris (Amsterdam, 1692), p. 156.
5 J. Ozanam, Traite des lignes du premier genre (Paris, 1687), p. 8; Ozanam,
Count de mathemalique, Tome III (Paris, 1693), p. 139.
6 A. dc Moivre in Philosophical Transactions, Vol. XIX (London, 1698), p. 52;
De Moivre, Miscellanea analylica de seriebus (London, 1730), p. 235.
7 David Gregory, Ada eruditorum (1703), p. 456.
8 L'Abbe* Deidier, La Mesure des Surfaces el des Solides (Paris, 1740), p. 181.
9 B. F. de Belidor, op. tit. (Paris, 1725), p. 70.
10 Maria G. Agnesi, Inslituzioni analitiche, Tome I (Milano, 1748), p. 76.
PROPORTION 293
Nicolaas Ypey,1 and Manfredi.2 This use of the comma for ratio,
rather than the Oughtredian dot, does not seem to be due to any
special cause, other than the general tendency observable also in the
notation for decimal fractions, for writers to use the dot and comma
more or less interchangeably.
An odd designation occurs in an English edition of Ozanam,3
namely, "A .2.5.3:: CA.D.6," where A,£,C,Z> are quantities in
geometrical proportion and the numbers are thrown in by way of
concrete illustration.
258. The notation: :: : in Europe and America. — The colon which
replaced the dot as the symbol for ratio was slow in making its appear-
ance on the Continent. It took this symbol about half a century to
cross the British Channel. Introduced in England by Vincent Wing
in 1651, its invasion of the Continent hardly began before the begin-
ning of the eighteenth century. We find the notation A:B::C:D
used by Leibniz,4 Johann Bernoulli,5 De la Hire,6 Parent,7 Bomie,8
Saulmon,9 Swedcnborg,10 Lagny,11 Senes,12 Chevalier de Louville,13
Clairaut,14 Bouguer,15 Nicole (1737, who in 1715 had used . :: .),16La
1 Nicolaas Ypey, Grondbeginselen der Keegelsneeden (Amsterdam, 1769),
p. 3.
2 Gabriello Manfredi, De Constructione Aequationum differentialium primi
(jradus (1707), p. 123.
3 J. Ozanam, Cursus mathematicus, translated "by several Hands" (London,
1712), Vol. I, p. 199.
4 Ada crudiiorum (1684), p. 472.
6 Johanne (I) Bernoulli in Journal des S^avans, ann6e 1698 (Amsterdam,
1709), p. 576. See this notation used also in Tann6e 1791 (Amsterdam, 1702),
p. 371.
6 De la Hire in Histoire de Vacad6mie r. des sciences, ann6e 1708 (Paris, 1730),
"Memoires," p. 57.
7 A. Parent in op. cit., ann<5e 1712 (Paris, 1731), "M6moires," p. 98.
8 Bomic in op. cit., p. 213.
9 Saulmon in op. cit., p. 283.
10 Emanuel Swcdberg, Daedalus Hyperboreus (Upsala, 1716).
11 T. F. Lagny in Histoire de I'academie r. des sciences, ann6e 1719 (Paris, 1721),
"Memoires," p. 139.
12 Dominique de Senes in op. cit., p. 363.
13 De Louville in op. cit., ami6e 1724 (Paris, 1726), p. 67.
14 Clairaut in op. cit., ann6e 1731 (Paris, 1733), "M6moires," p. 484.
16 Pierre Bougver in op. cit., anne*e 1733 (Paris, 1735), "Me"moires," p. 89.
16 F. Nicole in op. cit., ann6e 1737 (Paris, 1740), "Me*moires," p. 64.
294 A HISTORY OF MATHEMATICAL NOTATIONS
Caille,1 D'Alembert,2 Vicenti Riccati,3 and Jean Bernoulli.4 In the
Latin edition of De la Caille's5 Lectiones four notations are explained,
namely, 3.12::2.8, 3:12::2:8, 3:12 = 2:8, 3|12||2|8, but the nota-
tion 3: 12:: 2:8 is the one actually adopted.
The notation : : : : was commonly used in England and the United
States until the beginning of the twentieth century, and even now in
those countries has not fully surrendered its place to : = : . As late
as 1921 : :: : retains its place in Edwards' Trigonometry* and it occurs
in even later publications. The : : : : gained full ascendancy in Spain
and Portugal, and in the Latin- American countries. Thus it was used
in Madrid by Juan Gerard,7 in Lisbon by Joao Felix Pereira8 and
Luiz Porfirio da Motta Pegado,9 in Rio de Janeiro in Brazil by Fran-
cisco Miguel Pires10 and C. B. Ottoni,11 at Lima in Peru by Maximo
Vazquez12 and Luis Monsante,13 at Buenos Ayres by Florentino Garcia,14
at Santiago de Chile by Gabriel Izquierdo,15 at Bogota in Colombia
by Indalecio Li6vano,16 at Mexico by Juan Maria Balbontin.17
1 La Caille in op. ciL, annee 1741 (Paris, 1744), p. 256.
2 D'Alembert in op. ciL, ann<5c 1745 (Paris, 1749), p. 367.
3 Vincent! Riccati, Opusculorum ad res physicas et mathcmaticas pertinentium.
Tomus primus (Bologna, 1757), p. 5.
4 Jean Bernoulli in Nouveaux memoires de Vacadernie r. des sciences et belles-
lettres, ann6c 1771 (Berlin, 1773), p. 286.
6 N. L. de la Caille, Lectiones elementares malhematicae , ... in Latinum tra-
ductae et ad editionem Parisinam anni MDCCL1X denuo cxactae a C [arolo]
S [cherffer] e S. J. (Vienna, 1762), p. 76.
6R. W. K. Edwards, An Elementary Text-Book of Trigonometry (new ed.;
London, 1921), p. 152.
7 Juan Gerard, Presbitero, Tratado completo de aritmetica (Madrid, 1798), p. 69.
8 J. F. Pereira, Rudimentos de arithmetica (QuartaEdigao; Lisbon, 1863), p. 129.
9 Luiz Porfirio da Motta Pegado, Tratado elementar de arithmetica (Secunda
edigao; Lisbon, 1875), p. 235.
10 Francisco Miguel Pires, Tratado de Trigonometria Espherica (Rio de Janeiro,
1866), p. 8.
11 C. B. Ottoni, Elementos de geometria e trigonometria reclilinea (4th ed.;
Rio de Janeiro, 1874), "Trigon.," p. 36.
12 Maximo Vazquez, Aritmetica practica (7th cd.; Lima, 1875), p. 130.
13 Luis Monsante, Lecciones de aritmetica demostrada (7th ed.; Lima, 1872),
p. 171.
14 Florentino Garcia, El aritmetica Argentino (5th ed.; Buenos Aires, 1871),
p. 41; first edition, 1833.
16 Gabriel Izquierdo, Tratado de aritmetica (Santiago, 1859), p. 157.
16 Indalecio LieVano, Tratado de aritmetica (2d ed.; Bogota, 1872), p. 148.
17 Juan Maria Balbontin, Tratado elemental de aritmetica (Mexico, 1878), p. 96.
PROPORTION 295
259. The notation of Leibniz. — In the second half of the eighteenth
century this notation, A:B::C:D, had gained complete ascendancy
over A.B::C.D in nearly all parts of Continental Europe, but at
that very time it itself encountered a serious rival in the superior
Leibnizian notation, A:B = C:D. If a proportion expresses the
equality of ratios, why should the regular accepted equality sign
not be thus extended in its application? This extension of the sign
of equality = to writing proportions had already been made by
Stampioen (§ 256). Leibniz introduced the colon (:) for ratio and for
division in the Ada eruditorum of 1684, page 470 (§ 537). In 1693
Leibniz expressed his disapproval of the use of special symbols for ratio
and proportion, for the simple reason that the signs for division and
equality are quite sufficient. He1 says: "Many indicate by a-r-b^rc + d
that the ratios a to b and c to d are alike. But I have always disap-
proved of the fact that special signs are used in ratio and proportion,
on the ground that for ratio the sign of division suffices and likewise
for proportion the sign of equality suffices. Accordingly, I write the
ratio a to & thus: a:b or j- just as is done in dividing a by b. I desig-
nate proportion, or the equality of two ratios by the equality of the
two divisions or fractions. Thus when I express that the ratio a to 6
CL C
is the same as that of c to d, it is sufficient to write a:b = c:d or r =-3 ."
6 a
Cogent as these reasons are, more than a century passed be-
fore his symbolism for ratio and proportion triumphed over its
rivals.
Leibniz 's notation, a:b = c:d, is used in the Ada eruditorum of
1708, page 271. In that volume (p. 271) is laid the editorial policy
that in algebra the Leibnizian symbols shall be used in the Ada. We
quote the following relating to division and proportion (§197):
"We shall designate division by two dots, unless circumstance should
prompt adherence to the common practice. Thus, we shall have
a:6 = r. Hence with us there will be no need of special symbols for
denoting proportion. For instance, if a is to b as c is to d, we have
a:b = c:d."
The earliest influential textbook writer who adopted Leibniz'
notation was Christian Wolf. As previously seen (§ 255) he sometimes
1 G. W. Leibniz, Matheseos universalis pars prior, de Terrainis incomplexis,
No. 16; reprinted in Gesammelte Werke (C. I. Gerhardt), 3. Folge, II3, Band VII
(Halle, 1863), p. 56.
296 A HISTORY OF MATHEMATICAL NOTATIONS
wrote a.b^c.d. In 17101 he used both 3.12::5.20 and 3:12 = 5:20,
but from 17132 on, the Leibnizian notation is used exclusively.
One of the early appearances of a : b = c : d in France is in Clairaut's
algebra3 and in Saverien's dictionary,4 where Saverien argues that the
equality of ratios is best indicated by = and that :: is superfluous.
It is found in the publications of the Paris Academy for the year 1765,5
in connection with Euler who as early as 1727 had used it in the com-
mentaries of the Petrograd Academy.
Benjamin Panchaud brought out a text in Switzerland in 1743,6
using : = :. In the Netherlands7 it appeared in 1763 and again in
1775.8 A mixture of Oughtred's symbol for ratio and the = is seen in
Pieter Venema9 who writes . = .
In Vienna, Paulus Mako10 used Leibniz' notation both for geo-
metric and arithmetic proportion. The Italian Petro Giannini11 used
: = : for geometric proportion, as does also Paul Frisi.12 The first volume
of Ada Helvetia13 gives this symbolism. In Ireland, Joseph Fenn14 used
it about 1770. A French edition of Thomas Simpson's geometry15
uses : = : . Nicolas Fuss16 employed it in St. Petersburgh. In England,
1 Chr. Wolf, Anfangsgninde alter mathematischen Wissenschaften (Magdeburg,
1710), Vol. I, p. 65. See J. Tropfke, Geschichte der EUmentar-Mathematik, Vol. Ill
(2d ed.; Berlin and Leipzig, 1922), p. 14.
2 Chr. Wolf, Elementa malhescos universae, Vol. I (Halle, 1713), p. 31.
3 A, C. Clairaut, Siemens d'algebre (Paris, 1746), p. 21.
4 A. Saverien, Diclionnaire universel de mathematique et physique (Paris, 1753),
arts. "Raisons semblables," "Caractere."
5 Histoire de V academic r. des sciences, anne*e 1765 (Paris, 1768), p. 563;
Commentarii academiae scientiarum .... ad annum 1727 (Petropoli, 1728), p. 14.
6 Benjamin Panchaud, Entretiens on lemons mathtmatiques (Lausanne, Geneve,
1743), p. 226.
7 A. R. Maudvit, Inleiding tot de Keegel-Sneeden (Shaage, 1763).
8 J. A. Fas, Inleiding tot de Kennisse en het Gebruyk der Oneindig Kleinen (Ley-
den, 1775), p. 80.
9 Pieter Venema, Algebra ofte Stel-Konst, Vierde Druk (Amsterdam, 1768),
p. 118.
10 Pavlvs Mako, Compendiaria matheseos institutio (editio altera; Vindobonae,
1766), p. 169, 170.
11 Petro Giannini, Opuscola mathemaiica (Parma, 1773), p. 74.
12 Paulli Frisii Operum, Tomus Secundus (Milan, 1783), p. 284.
18 Ada Helvetica, physico-mathematico-Botanico-Medica, Vol. I (Basel, 1751),
p. 87.
14 Joseph Fenn, The Complete Accountant (Dublin, [n.d.]), p. 105, 128.
15 Thomas Simpson, Element de geometric (Paris, 1766).
16 Nicolas Fuss, Lemons de geomttrie (St. PStersbourg, 1798), p. 112.
EQUALITY 297
John Cole1 adopted it in 1812, but a century passed after this date
before it became popular there.
The Leibnizian notation was generally adopted in Europe during
the nineteenth century.
In the United States the notation : :: : was the prevailing one
during the nineteenth century. The Leibnizian signs appeared only
in a few works, such as the geometries of William Chauvenet2 and
Benjamin Peirce.3 It is in the twentieth century that the notation
: = : came to be generally adopted in the United States.
A special symbol for variation sometimes encountered in English
and American texts is oc , introduced by Emerson.4 "To the Common
Algebraic Characters already received I add this oc , which signifies a
BC
general Proportion; thus, -4.ee--—, signifies that A is in a constant
RC1
ratio to -jr- ." The sign was adopted by Chrystal,5 Castle,6 and others.
SIGNS OP EQUALITY
260. Early symbols. — A symbol signifying "it gives" and ranking
practically as a mark for equality is found in the linear equation of the
Egyptian Ahmcs papyrus (§ 23, Fig. 7). We have seen (§ 103) that
Diophantus had a regular sign for equality, that the contraction pha
answered that purpose in the Bakhshal! arithmetic (§ 109), that the
Arab al-Qalasadi used a sign (§ 124), that the dash was used for the
expression of equality by Regiomontanus (§ 126), Pacioli (§ 138),
and that sometimes Cardan (§ 140) left a blank space where we would
place a sign of equality.
261. Recorders sign of equality. — In the printed books before
Recordc, equality was usually expressed rhetorically by such words
as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and some-
times by the abbreviated form aeq. Prominent among the authors
expressing equality in some such manner are Kepler, Galileo, Torri-
celli, Cavalieri, Pascal, Napier, Briggs, Gregory St. Vincent, Tacquet,
and Fermat. Thus, about one hundred years after Recorde, some of
1 John Cole, Stereogoniometry (London, 1812), p. 44, 265.
2 William Chauvenet, Treatise on Elementary Geometry (Philadelphia, 1872),
p. 69.
3 Benjamin Peirce, Elementary Treatise on Plane and Solid Geometry (Boston,
1873), p. xvi.
4 W. Emerson, Doctrine of Fluxions (3d ed.; London, 1768), p. 4.
s G. Chrystal, Algebra, Part I, p. 275.
6 Frank Castle, Practical Mathematics for Beginners (London, 1905), p, 317.
298 A HISTORY OF MATHEMATICAL NOTATIONS
the most noted mathematicians used no symbol whatever for the
expression of equality. This is the more surprising if we remember
that about a century before Recorde, Regiomontanus (§ 126) in his
correspondence had sometimes used for equality a horizontal dash — ,
that the dash had been employed also by Pacioli (§ 138) and Ghaligai
(§139). Equally surprising is the fact that apparently about the time
of Recorde a mathematician at Bologna should independently origi-
nate the same symbol (Fig. 53) and use it in his manuscripts.
Recorders = , after its ddbut in 1557, did not again appear in
print until 1618, or sixty-one years later. That some writers used
symbols in their private manuscripts which they did not exhibit in
their printed books is evident, not only from the practice of Regio-
montanus, but also from that of John Napier who used Recorders =
in an algebraic manuscript which he did not publish and which was
first printed in 1839.1 In 1618 we find the = in an anonymous Appen-
dix (very probably due to Oughtred) printed in Edward Wright's
English translation of Napier's famous Descriptio. But it was in
1631 that it received more than general recognition in England by
being adopted as the symbol for equality in three influential works,
Thomas Harriot's Artis analyticae praxis, William Oughtred's Claris
mathematicae, and Richard Norwood's Trigonometria.
262. Different meanings of =. — As a source of real danger to
Recorde's sign was the confusion of symbols which was threatened on
the European Continent by the use of = to designate relations other
than that of equality. In 1591 Francis Vieta in his In artem analyticen
isagoge used = to designate arithmetical difference (§ 177). This
designation was adopted by Girard (§ 164), by Sieur de Var-Lezard2
in a translation of Vieta's Isagoge from the Latin into French, De
Graaf,3 and by Franciscus a Schooten4 in his edition of Descartes'
Geometrie. Descartes5 in 1638 used = to designate plus ou mains,
i.e., ±.
Another complication arose from the employment of = by Johann
1 Johannis Napier, De Arte Logistica (Edinburgh, 1839), p. 160.
2 1. L. Sieur de Var-Lezard, Introduction en I'art analytic ov nouvelle algebre de
Francois Viete (Paris, 1630), p. 36.
8 Abraham de Graaf, De beginselen van de Algebra of Stelkonst (Amsterdam,
1672), p. 26.
* Renati Descartes, Geometria (ed. Franc, a Schooten; Francofvrti al Moenvm,
1695), p. 395.
* (Euvres de Descartes (eU Adam et Tannery), Vol. II (Paris, 1898), p. 314,
426.
EQUALITY 299
Caramuel1 as the separatrix in decimal fractions; with him 102 = 857
meant our 102.857. As late as 1706 G. H. Paricius2 used the signs
= , :, and — as general signs to separate numbers occurring in the
process of solving arithmetical problems. The confusion of algebraic
language was further increased when Dulaurens3 and Reyher4 desig-
nated parallel lines by = . Thus the symbol = acquired five different
meanings among different continental writers. For this reason it was
in danger of being discarded altogether in favor of some symbol which
did not labor under such a handicap.
263. Competing symbols. — A still greater source of danger to our
= arose from competing symbols. Pretenders sprang up early on
both the Continent and in England. In 1559, or two years after the
appearance of Recorders algebra, the French monk, J. Buteo,6 pub-
lished his Logistica in which there appear equations like "LA, -J-B, %C
[14" and "3A .3#. 15C[120," which in modern notation are x+$y+$z
= 14 and 3x+3?/+ 15-2 = 120. Buteo's [ functions as a sign of equality.
In 1571, a German writer, Wilhelm Holzmann, better known under
the name of Xylander, brought out an edition of Diophantus' Arith-
metica* in which two parallel vertical lines || were used for equality.
He gives no clue to the origin of the symbol. Moritz Cantor7 suggests
that perhaps the Greek word iaoi ("equal") was abbreviated in the
manuscript used by Xylander, by the writing of only the two letters
u. Weight is given to this suggestion in a Parisian manuscript on
Diophantus where a single i denoted equality.8 In 1613, the Italian
writer Giovanni Camillo Glorioso used Xylander's two vertical lines
for equality.9 It was used again by the Cardinal Michaelangelo
Ricci.10 This character was adopted by a few Dutch and French
1 Joannis Caramuelis, Mathesis Biceps veins et nova (1670), p. 7.
2 Georg Heinrich Paricius, Praxis arithmelices (Regensburg, 1706). Quoted
by M. Sterner, Geschichte der Rechenkunst (Munchen und Leipzig, 1891), p. 348.
3 Franc. ois Dulaurens, Specimina mathemalica (Paris, 1667).
4 Samuel Reyher, Euclides (Kiei, 1698).
8 J. Buteo, Logistica (Leyden, 1559), p. 190, 191. See J. Tropfke, op. cit.,
Vol. Ill (2d ed.; Leipzig, 1922), p. 136.
6 See Nesselmann, Algebra der Griechen (1842), p. 279.
7 M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. II (2d ed.;
Leipzig, 1913), p. 552.
8 M. Cantor, op. cit., Vol. I (3d ed.; 1907), p. 472.
9 Joannis Camillo Gloriosit Ad theorema geometricvm (Venetiis, 1613), p. 26.
10 Michaelis Angeli Riccii, Exercitatio geometrica de maximis el minimis (Lon-
dini, 1668), p. 9.
300 * A HISTORY OF MATHEMATICAL NOTATIONS
mathematicians during the hundred years that followed, especially
in the writing of proportion. Thus, R. Descartes,1 in his Opuscules
de 1619-1621 , made the statement, "ex progressione 1 1 2 1 1 4 1 8 1 1 16 1 32 1|
habentur numeri perfecti 6, 28, 496." Pierre de Carcavi, of Lyons, in
a letter to Descartes (Sept. 24, 1649), writes the equation "+1296—
3060a+2664a2-1115aM-239a4-25a5+a6 1| 0," where "la lettre a est
Pinconnue en la rnaniere de Monsieur Vieta" and || is the sign of
equality.2 De Monconys3 used it in 1666; De Sluse4 in 1668 writes
our be=a? in this manner "be \\ aa." De la Hire (§254) in 1701 wrote
the proportion a:b = x2:ab thus: "a|6||o;x|ab." This symbolism is
adopted by the Dutch Abraham de Graaf5 in 1703, by the Frenchman
Parent6 in 1713, and by certain other writers in the Journal des
S$avans.7 Though used by occasional writers for more than a century,
this mark || never gave promise of becoming a universal symbol for
equality. A single vertical line was used for equality by S. Ileyher
in 1698. With him, "A\B" meant A = #. He attributes8 this notation
to the Dutch orientalist and astronomer Jacob Golius, saying: "Espe-
cially indebted am I to Mr. Golio for the clear algebraic mode of dem-
onstration with the sign of equality, namely the rectilinear stroke
standing vertically between two magnitudes of equal measure."
In England it was Leonard and Thomas Digges, father and son,
who introduced new symbols, including a line complex X for equality
(Fig. 78).'
The greatest oddity was produced by H6rigone in his Cursus
mathematicus (Paris, 1644; 1st ed., 1634). It was the symbol "2|2."
Based on the same idea is his "3|2" for "greater than," and his "2|3"
for "less than." Thus, a?+ab = b2 is indicated in his symbolism by
1 (Euvrcs de Descartes, Vol. X (1908), p. 241.
2 Op. cit., Vol. V (1903), p. 418.
3 Journal des voyages de Monsieur de Monconys (Troisie'me partie ; Lyon,
166G), p. 2. Quoted by Henry in Revue archeologique (N.S.), Vol. XXXVII (1879),
p. 333.
4 Renati Francisci Slusii Mesolabum, Leodii Eburonum (1668), p. 51.
5 Abraham de Graaf, De Vervulling van de Geometria en Algebra (Amsterdam,
1708), p. 97.
8 A. Parent, Essais el recherches de mathematique et de physique (Paris, 1713),
p. 224.
7 Journal des Sc.avans (Amsterdam, for 1713), p. 140; ibid, (for 1715), p. 537;
and other years.
8 Samuel Ileyher, op. cit., Vorrede.
9 Thomas Digges, Stralioticos (1590), p. 35.
EQUALITY 301
"a2+6a2|262." Though clever and curious, this notation did not
appeal. In some cases H£rigone used also LJ to express equality. If
this sign is turned over, from top to bottom, we have the one used by
F. Dulaurens1 in 1667, namely, n ; with Dulaurens P signifies "majus,"
H signifies "minus"; Leibniz, in some of his correspondence and
unpublished papers, used2 n and also3 = ; on one occasion he used
the Cartesian4 » for identity. But in papers which he printed, only
the sign = occurs for equality.
Different yet was the equality sign 3 used by J. V. Andrea5 in
1614.
The substitutes advanced by Xylander, Andrea, the two Digges,
Dulaurens, and Herigone at no time seriously threatened to bring
about the rejection of Recorders symbol. The real competitor was the
mark » , prominently introduced by Ren6 Descartes in his Geometric
(Leyden, 1637), though first used by him at an earlier date.6
264. Descartes1 sign of equality. — It has been stated that the sign
was suggested by the appearance of the combined ae in the word
aequalis, meaning "equal." The symbol has been described by Cantor7
as the union of the two letters ae. Better, perhaps, is the description
given by Wieleitner8 who calls it a union of oe reversed; his minute
examination of the symbol as it occurs in the 1637 edition of the
Geometric revealed that not all of the parts of the letter e in the
combination oe are retained, that a more accurate way of describing
that symbol is to say that it is made up of two letters o, that is, oo
pressed against each other and the left part of the first excised. In
some of the later appearances of the symbol, as given, for example,
by van Schooten in 1659, the letter e in oe, reversed, remains intact.
We incline to the opinion that Descartes' symbol for equality, as it
appears in his Geometric of 1637, is simply the astronomical symbol
1 F. Dulaurens, Specimina mathemalica (Paris, 1667).
2 C. I. Gerhardt, Leibnizens mathematische Schriften, Vol. I, p. 100, 101, 155,
163, etc.
3 Op. dt., Vol. I, p. 29, 49, 115, etc.
4 Op. ciL, Vol. V, p. 150.
5 Joannis Valentini Andreae, Collectaneorum Mathematicorum decades XI
(Tubirigae, 1614). Taken from P. Treutlein, "Die deutsche Coss," Abhandlungen
zur Geschichte der Mathematik, Vol. II (1879), p. 60.
* (Entires de Descartes (ed. Ch. Adam et P. Tannery), Vol. X (Paris, 1908),
p. 292, 299.
7 M. Cantor, op. cit., Vol. II (2d ed., 1913), p. 794.
8 II. Wieleitner in Zeitschr. fur math. u. naiuriviss. Unterricht, Vol. XL VII
(1916), p. 414.
302 A HISTORY OF MATHEMATICAL NOTATIONS
for Taurus, placed sideways, with the opening turned to the left.
This symbol occurs regularly in astronomical works and was there-
fore available in some of the printing offices.
Descartes does not mention Recorde's notation; his Geometrie is
void of all bibliographical and historical references. But we know that
he had seen Harriot's Praxis, where the symbol is employed regularly.
In fact, Descartes himself1 used the sign = for equality in a letter of
1640, where he wrote "1C-6# = 40" for 0^-60; = 40. Descartes does
not give any reason for advancing his new symbol *> . We surmise that
Vieta's, Girard's, and De Var-Lezard's use of = to denote arith-
metical "difference" operated against his adoption of Recorde's sign.
Several forces conspired to add momentum to Descartes' symbol » .
In the first place, the Geometrie, in which it first appeared in print,
came to be recognized as a work of genius, giving to the world analytic
geometry, and therefore challenging the attention of mathematicians.
In the second place, in this book Descartes had perfected the expo-
nential notation, an (n, a positive integer), which in itself marked a
tremendous advance in symbolic algebra; Descartes' » was likely to
follow in the wake of the exponential notation. The » was used by
F. Debeaune2 as early as October 10, 1638, in a letter to Roberval.
As Descartes had lived in Holland several years before the appear-
ance of his Geometrie, it is not surprising that Dutch writers should be
the first to adopt widely the new notation. Van Schooten used the
Cartesian sign of equality in 1646.3 He used it again in his translation
of Descartes' Geometrie into Latin (1649), and also in the editions of
1659 and 1695. In 1657 van Schooten employed it in a third publica-
tion.4 Still more influential was Christiaan Huygens5 who used » as
early as 1646 and in his subsequent writings. He persisted in this
usage, notwithstanding his familiarity with Recorde's symbol through
the letters he received from Wallis and Brounckcr, in which it occurs
many times.6 The Descartian sign occurs in the writings of Hudde
and De Witt, printed in van Schooten's 1659 and later editions of
Descartes' Geometrie. Thus, in Holland, the symbol was adopted by
1 (Euvres dc Descartes, Vol. Ill (1899), p. 190.
2 Ibid., Vol. V (1903), p. 519.
3 Francisci i\ Schooten, DC organica conicarum sectionum (Leyden, 1646), p. 91.
4 Francisci & Schooten, Exercitationvm malhematicarum liber primus (Lcyden,
1657), p. 251.
6 (Euvres completes de Christiaan Huygens , Tome I (La Haye, 1888), p. 26, 526.
6 Op. cit., Tome II, p. 296, 519; Tome IV, p. 47, 88.
EQUALITY 303
the most influential mathematicians of the seventeenth century. It
worked its way into more elementary textbooks. Jean Prestet1
adopted it in his Nouveaux Elemens, published at Paris in 1689. This
fact is the more remarkable, as in 1675 he2 had used the sign = . It
seems to indicate that soon after 1675 the sign » was gaining over =
in France. Ozanam used » in his Didionaire mathematique (Amster-
dam, 1691), though in other books of about the same period he used
^, as we see later. The Cartesian sign occurs in a French text by
Bernard Lamy.3
In 1659 Descartes' equality symbol invaded England, appearing
in the Latin passages of Samuel Foster's Miscellanies. Many of the
Latin passages in that volume are given also in English translation.
In the English version the sign = is used. Another London publica-
tion employing Descartes' sign of equality was the Latin translation
of the algebra of the Swiss Johann Alexander.4 Michael Rolle uses »
in his Traite d'algebre of 1690, but changes to = in 1709.5 In Hol-
land, Descartes' equality sign was adopted in 1660 by Kinckhvysen,6
in 1694 by De Graaf,7 except in writing proportions, when he uses =.
Bernard Nieuwentiit uses Descartes' symbol in his Considerations of
1694 and 1696, but preferred = in his Analysis infinitorum of 1695.
De la Hire8 in 1701 used the Descartian character, as did also Jacob
Bernoulli in his Ars Conjectandi (Basel, 1713). Descartes' sign of
equality was widely used in France and Holland during the latter part
of the seventeenth and the early part of the eighteenth centuries, but
it never attained a substantial foothold in other countries.
265. Variations in the form of Descartes1 symbol. — Certain varia-
tions of Descartes' symbol of equality, which appeared in a few texts,
are probably due to the particular kind of symbols available or irn-
provisable in certain printing establishments. Thus Johaan Cara-
1 Jean Prestet, Nouveaux Siemens des mathemaliques, Vol. I (Paris, 1689),
p. 261.
2 J. P. [restet] Siemens des mathtmaliques (Paris, 1675), p. 10.
3 Bernard Lamy, Elemens des mathemaliques (3d ed.; Amsterdam, 1692), p. 93.
4 Synopsis Algebraica, Opus posthumum Johannis Alexandri, Bernalis-Helvetii.
In usurn scholae malhematicae apud Hospitium-Christi Londinense (Londirii, 1693),
p. 2.
5 Mem. de Vacademie royale des sciences, anne*e 1709 (Paris), p. 321.
6 Gerard Kinckhvysen, De Grondt der Meet-Konst (Te Haerlem, 1660), p. 4.
7 Abraham de Graaf, De Geheele Mathesis of Wiskonst (Amsterdam, 1694),
p. 45.
8 De la Hire, Nouveaux tttmens des sections coniques (Paris, 1701), p. 184.
304 A HISTORY OF MATHEMATICAL NOTATIONS
muel1 in 1670 employed the symbol 7E; the 1679 edition of Format's2
works gives oo in the treatise Ad locos pianos et solidos isagoge, but in
Fermat's original manuscripts this character is not found.8 On the
margins of the pages of the 1679 edition occur also expressions of
which "DA {BE" is an example, where DA=BE. J. Ozanam4 em-
ploys V^N in 1682 and again in 1693; he refers to £ as used to mark
equality, "mais nous le changerons en celuy-cy, oo ; que nous semble
plus propre, et plus naturel." Andreas Spole5 said in 1692: u~ vel =
est nota aequalitates." Wolff6 gives the Cartesian symbol inverted,
thus a .
266. Struggle for supremacy. — In the seventeenth century,
Recorders = gained complete ascendancy in England. We have seen
its great rival » in only two books printed in England. After Harriot
and Oughtred, Recorders symbol was used by John Wallis, Isaac
Barrow, and Isaac Newton. No doubt these great names helped the
symbol on its way into Europe.
On the European Continent the sign = made no substantial
headway until 1650 or 1660, or about a hundred years after the appear-
ance of Recorders algebra. When it did acquire a foothold there, it
experienced sharp competition with other symbols for half a century
before it fully established itself. The beginning of the eighteenth
century may be designated roughly as the time when all competition
of other symbols practically ceased. Descartes himself used = in a
letter of September 30, 1640, to Mersenne. A Dutch algebra of 1639
and a tract of 1640, both by J. Stampioen,7 and the Teutsche Algebra
of the Swiss Johann Heinrich Rahn (1659), are the first continental
textbooks that we have seen which use the symbol. Rahn says, p. 18:
"Bey disem anlaasz habe ich das namhafte gleichzeichen = zum
ersten gebraucht, bedeutet ist gleich, 2a = 4 heisset 2a ist gleich 4." It
was used by Bernhard Frenicle de Bessy, of magic-squares fame, in a
1 J. Caramuel, op. cit., p. 122.
2 Varia opera mathematica D. Petri de Fermai (Tolosae, 1679), p. 3, 4, 5.
8 (Euvres de Fermat (ed. P. Tannery et C. Henry), Vol. I (Paris, 1891), p. 91.
4 Journal des Sgavans (de Tan 1682), p. 160; Jacques Ozanam, Cours de Mathe-
matiques, Tome I (Paris, 1692), p. 27; also Tome III (Paris, 1693), p. 241.
6 Andreas Spole, Arithmetica vulgaris et specioza (Upsaliae, 1692), p. 16. See
G. Enestrom in U Intermediaire des mathematiciens, Tome IV (1897), p. 60.
6 Christian Wolff, Maihemalisches Lexicon (Leipzig, 1716), "Signa," p. 1264.
7 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel ('s Graven-Hage,
1639); J. Stampioenii Wisk-Konstich ende Reden-maetich Bewijs (s'Graven-IIage,
1640).
EQUALITY 305
letter1 to John Wallis of December 20, 1661, and by Huips2 in the
same year. Leibniz, who had read Barrow's Euclid of 1655, adopted
the Recordean symbol, in his De arte combinatoria of 1666 (§ 545), but
then abandoned it for nearly twenty years. The earliest textbook
brought out in Paris that we have seen using this sign is that of
Arnauld3 in 1667; the earliest in Leyden is that of C. F. M. Dechales4
in 1674.
The sign = was used by Prestet,5 Abbe Catelan and Tschirnhaus,6
Hoste,7 Ozanam,8 Nieuwentijt,9 Wcigel,10 De Lagny,11 Carre,12 L'Hospi-
tal,13 Polynier,14 Guisnee,15 and Reyneau.16
This list constitutes an imposing array of names, yet the majority
of writers of the seventeenth century on the Continent either used
Descartes' notation for equality or none at all.
267. With the opening of the eighteenth century the sign =
gained rapidly; James Bernoulli's Ars Conjectandi (1713), a post-
humous publication, stands alone among mathematical works of
prominence of that late date, using » . The dominating mathematical
advance of the time was the invention of the differential and integral
calculus. The fact that both Newton and Leibniz used Recorde's
symbol led to its general adoption. Had Leibniz favored Descartes'
I (Euvres completes des Christiaan-Huygens (Lallayc), Tome IV (1891), p. 45.
2Frans van der Huips, Algebra ofte een Noodige (Amsterdam, 1601), p. 178.
Reference supplied by L. C. Karpinski.
3 Antoine Arnauld, Nouveaux Siemens de Geometric (Paris, 1667; 2d ed.,
1683).
4 C. F. Dechales, Cvrsvs sev Mvndvs Mathematics, Tomvs tertivs (Lvgdvni,
1674), p. 666; Editio altera, 1690.
5 J. P[restet], op. ciL (Paris, 1675), p. 10.
6 Ada eruditorum (anno 1682), p. 87, 393.
7 P. Hoste, Recueil des traites de mathematiques, Tome III (Paris, 1692), p. 93.
8 Jacques Ozanam, op. cit., Tome I (nouvelle 6d.; Paris, 1692), p. 27. In
various publications between the dates 1682 and 1693 Ozanam used as equality
signs / — ', », and =.
9 Bernard Nieuwentijt, Analysis infinilorum.
10 Erhardi Wvigelii Philosophia mathematica (Jcnae, 1693), p. 135.
II Thomas F. de Lagny, Nouveaux eUrnens d'arithrnetique, et d'algebre (Paris,
1697), p. 232.
12 Louis Carre*, Methode pour la mesure des surfaces (Paris, 1700), p. 4.
13 Marquis de I'Hospital, Analyse des Infmiment Petits (Paris, 1696, 1715).
14 Pierre Polynier, EUmens des Mathematiques (Paris, 1704), p. 3.
16 Guisnee, Application de Valgebre a la geometric (Paris, 1705).
16 Charles Reyneau, Analyse demontree, Tome I (1708).
306 A HISTORY OF MATHEMATICAL NOTATIONS
» , then Germany and the rest of Europe would probably have joined
France and the Netherlands in the use of it, and Recorded symbol
would probably have been superseded in England by that of Descartes
at the time when the calculus notation of Leibniz displaced that of
Newton in England. The final victory of = over » seems mainly
due to the influence of Leibniz during the critical period at the close of
the seventeenth century.
The sign of equality = ranks among the very few mathematical
symbols that have met with universal adoption. Recorde proposed
no other algebraic symbol; but this one was so admirably chosen that
it survived all competitors. Such universality stands out the more
prominently when we remember that at the present time there is still
considerable diversity of usage in the group of symbols for the differ-
ential and integral calculus, for trigonometry, vector analysis, in fact,
for every branch of mathematics.
The difficulty of securing uniformity of notation is further illus-
trated by the performance of Peter van Musschenbroek,1 of Leyden,
an eighteenth-century author of a two-volume text on physics, widely
known in its day. In some places he uses = for equality and in others
for ratio; letting S. s. be distances, and T. t .times, he says: "Erit S. s.
:: T. t. exprimunt hoc Mathematici scribendo, est S= T. sive Spatium
est uti tempus, nam signum = non exprimit aequalitatem, sed ratio-
nem." In writing proportions, the ratio is indicated sometimes by a
dot, and sometimes by a comma. In 1754, Musschenbroek had used
» for equality.2
268. Variations in the form of Recorde's symbol. — There has been
considerable diversity in the form of the sign of equality. Recorde
drew the two lines very long (Fig. 71) and close to each other, .
This form is found in Thomas Harriot's algebra (1631), and occa-
sionally in later works, as, for instance, in a paper of De Lagny3 and
in Schwab's edition of Euclid's Data.4 Other writers draw the two
lines very short, as does Weigel5 in 1693. At Upsala, Emanuel
van Musschenbroek, Introdttctio ad philosophiam naluralem, Vol. I
(Leyden, 1762), p. 75, 126.
2 Petri van Musschenbroek, Dissertaiio physica experimentalis de magnete
(Vienna), p. 239.
8De Lagny in Memvires de I'acadtmie r. d. sciences (depuis 1666 jusqu'a
1699), Vol. II (Paris, 1733), p. 4.
4 Johann Christoph Schwab, Eudida Data (Stuttgart, 1780), p. 7.
» Erhardi Weigeli PhUosophia mathematica (Jena, 1693), p. 181.
EQUALITY 307
Swedenborg1 makes them very short and slanting upward, thus //.
At times one encounters lines of moderate length, drawn far apart z ,
as in an article by Nicole2 and in other articles, in the Journal des
S$avans. Frequently the type used in printing the symbol is the figurfc
1, placed horizontally, thus3 ^ or4 £•
In an American arithmetic5 occurs, "1+6, = 7, X6 = 42, 4-2 = 21."
Wolfgang Bolyai6 in 1832 uses =z to signify absolute equality; 21,
equality in content; A(~B or B = )A,to signify that each value of A
is equal to some value of B; A( = )B, that each of the values of A is
equal to some value of B, and vice versa.
To mark the equality of vectors, Bellavitis7 used in 1832 and later
the sign =£=.
Some recent authors have found it expedient to assign = a more
general meaning. For example, Stolz and Grneiner8 in their theoretical
arithmetic write a o 6 = c and read it "a mit 6 ist c," the = signifying
"is explained by" or "is associated with." The small circle placed
between a and b means, in general, any relation or Verknupfung.
De Morgan9 used in one of his articles on logarithmic theory a
double sign of equality = = in expressions like (be0l/~1)x = = nelf^/~1J
where ft and v are angles made by 6 and n, respectively, with the initial
line. He uses this double sign to indicate "that every symbol shall
express not merely the length and direction of a line, but also the
quantity of revolution by which a line, setting out from the unit line,
is supposed to attain that direction."
1 Emanuel Swedberg, Daedalus Hyperboreus (Upsala, 1716), p. 39. See fac-
simile reproduction in Kungliga Vctenskaps Societelens i Upsala Tvdhundradrsminne
(Upsala, 1910).
2 Francois Nicole in Journal des Sgavans, Vol. LXXXIV (Amsterdam, 1728),
p. 293. See also anne'e 1690 (Amsterdam, 1691), p. 468; ann6e 1693 (Amsterdam,
1694), p. 632.
3 James Gregory, Geometria Pars Vniversalis (Padua, 1668); Emanuel Swed-
berg, op. cit., p. 43.
4 H. Vitalis, Lexicon mathematicum (Rome, 1690), art. "Algebra."
5 The Columbian Arithmetician, "by an American" (Haverhill [Mass.], 1811),
p. 149.
0 Wolfgang! Bolyai de Bolya, Tentamen (2ded.), Tom. I (Budapestini, 1897),
p. xi.
7 Guisto Bellavitis in Annali del R. Lomb.-Ven. (1832), Tom. II, p. 250-53.
8 0. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. I (2d ed.;
1911), p. 7.
9 A. de Morgan, Trans. Cambridge PhUos. Society, Vol. VII (1842), p. 186.
308 A HISTORY OF MATHEMATICAL NOTATIONS
269. Variations in the manner of using it. — A rather unusual use of
equality signs is found in a work of Deidier1 in 1740, viz.,
^ . ^j = ,
2+2+2 = 6"" 2 ; 4,4, 4, = 12~3+12*
H. Vitalis2 uses a modified symbol: "Nota =£= significat repctitam
aequationem . . . . vt 10 £6. "f~4^8~|~2." A discrimination between
= and » is made by Gallimard3 and a few other writers; " = , est
egale a; » qui signifie tout simplcmcnt, egal a , ou , qui est egal a."
A curious use, in the same expressions, of = , the comma, and the
word aequalis is found in a Tacquet-Whiston4 edition of Euclid, where
one reads, for example, "erit 8X432 = 3456 aequalis 8X400 = 3200,
+8X30 = 240, +8X2 = 16."
L. Gustave du Pasquier5 in discussing general complex numbers
employs the sign of double equality = to signify "equal by definition."
The relations between the coefficients of the powers of x in a series
may be expressed by a formal equality involving the series as a whole,
as in
where the symbol ^f indicates that the equality is only formal, not
arithmetical.6
270. Nearly equal. — Among the many uses made in recent years
of the sign ^> is that of "nearly equal to/' as in "e~\"] similarly, e^\
is allowed to stand for "equal or nearly equal to."7 A. Eucken8 lets r±:
stand for the lower limit, as in "J~45.10-40 (untere Grenze)," where J
means a mean moment of inertia. Greenhill9 denotes approximate
1 L'Abb6 Deidier, La mcsure des surfaces et des solides (Paris, 1740), p. 9.
2H. Vitalis, loc. cit.
3 J. E. Gallimard, La Science du calcul numerique, Vol. I (Paris, 1751), p. 3.
4 Andrea Tacquet, Elementa Euclidea geometriae [after] Guliclmus Whiston
(Amsterdam, 1725), p. 47.
5 Comptes Rendus du Congres International des Mathematicians (Strasbourg,
22-30 Septembre 1920), p. 164.
6 Art. "Algebra" in Encyclopaedia Britannica (1 1th ed., 1910).
7 A. Kratzer in Zeitschrift fur Physik, Vol. XVI (1923), p. 356, 357.
8 A. Eucken in Zeitschrifl der physikalischen Chernie, Band C, p. 159.
9 A. G. Greenhill, Applications of Elliptic Functions (London, 1892), p. 303,
340, 341.
COMMON FRACTIONS
309
equality by ^w- An early suggestion due to Fischer1 was the sign X
for "approximately equal to." This and three other symbols were
proposed by Boon2 who designed also four symbols for "greater than
but approximately equal to" and four symbols for "less than but
approximately equal to."
SIGNS OF COMMON FRACTIONS
271. Early forms. — In the Egyptian Ahmes papyrus unit fractions
were indicated by writing a special mark over the denominator
(§§22, 23). Unit fractions are not infrequently encountered among
the Greeks (§41), the Hindus and Arabs, in Leonardo of Pisa (§122),
and in writers of the later Middle Ages in Europe.3 In the text
Trisatika, written by the Hindu Sridhara, one finds examples like the
following: "How much money is there when half a kdkini, one-third
of this and one-fifth of this are added together?
Statement
1 1 1
1 2 3
Answer. Vardtikas 14."
This means lX-J + lXiXi+lX!X-i-Xi = 1\, and since 20 varatikas
= 1 kdkini, the answer is 14 varatikas.
John of Meurs (early fourteenth century)4 gives % as the sum of
three unit fractions -£, -j, and -g^, but writes "\ \ -J-," which is an
ascending continued fraction. He employs a slightly different nota-
tion for -gV, namely, "i -J - o \ o ."
Among Heron of Alexandria and some other Greek writers the
numerator of any fraction was written with an accent attached, and
was followed by the denominator marked with two accents (§41). In
some old manuscripts of Diophantus the denominator is placed above
the numerator (§ 104), and among the Byzantines the denominator
is found in the position of a modern exponent ;5 $ltt signified according-
ly A •
1 Ernst Gottfried Fischer, Lehrbuch der Elementar-Mathematik, 4. Theil,
Anfangsgrunde der Algebra (Berlin und Leipzig, 1829), p. 147. Reference given by
R. C. Archibald in Mathematical Gazette, Vol. VIII (London, 1917), p. 49.
2 C. F. Boon, Mathcinatical Gazette, Vol. VII (London, 1914), p. 48.
3 See G. Encstrom in Bibliotheca mathematica (3d ser.), Vol. XIV (1913-14),
p. 269, 270.
4 Vienna Codex 4770, the Quadripartitum numervrum, described by L. C.
Karpinski in Bibliotheca mathematics (3d scr.), Vol. XIII (1912-13), p, 109.
5 F. Hultsch, Metrologicorurn scriptorum reliquiae, Vol. I (Leipzig, 1864),
p. 173-75.
310 A HISTORY OF MATHEMATICAL NOTATIONS
The Hindus wrote the denominator beneath the numerator, but
without a separating line (§§ 106, 109, 113, 235).
In the so-called arithmetic of John of Seville,1 of the twelfth
century (?), which is a Latin elaboration of the arithmetic of al-
Khowarizmi, as also in a tract of Alnasavi (1030 A.D.),2 the Indian
mode of writing fractions is followed ; in the case of a mixed number,
the fractional part appears below the integral part. Alnasavi pur-
sues this course consistently3 by writing a zero when there is no intc-
«o Jy
gral part; for example, he writes ^ thus: i ,
272. The fractional line is referred to by the Arabic writer al-
Ha$$ar (§§ 122, 235, Vol. II § 422), and was regularly used by Leonardo
of Pisa (§§ 122, 235). The fractional line is absent in a twelfth-century
Munich manuscript;4 it was not used in the thirteenth-century writ-
ings of Jordanus Nemorarius,5 nor in the Gcrnardus algorithmus
demonstratus, edited by Joh. Schoner (Niirnberg, 1534), Part II,
chapter i.6 When numerator and denominator of a fraction are letters,
Gernardus usually adopted the form ab (a numerator, b denominator),
probably for graphic reasons. The fractional line is absent in the
Bamberger arithmetic of 1483, but occurs in Widman (1489), and in a
fifteenth-century manuscript at Vienna.7 While the fractional line
carne into general use in the sixteenth century, instances of its omis-
sion occur as late as the seventeenth century.
273. Among the sixteenth- and seventeenth-century writers
omitting the fractional line were Baeza8 in an arithmetic published at
Paris, Dibuadius9 of Denmark, and Paolo Casati.10 The line is
1 Boncompagni, Trattati d' aritmetica, VoL II, p. 16-72,
2H. Suter, Bibliotheca mathematica (3d ser.), Vol. VII (1906-7), p. 113-19.
3M. Cantor, op. cil.t VoL I (3d ed.), p. 762.
4 Munich MS Clm 13021. See Abhandlungen uber Geschichle der Mathematik,
Vol. VIII (1898), p. 12-13, 22-23, and the peculiar mode of operating with frac-
tions.
6 Bibliotheca malhematica (3d ser.), VoL XIV, p. 47.
6 Ibid., p. 143.
7 Codex Vindob. 3029, described by E. Rath in Bibliotheca mathematica (3d
ser.), VoL XIII (1912-13), p. 19. This manuscript, as well as Widman's arithmetic
of 1489, and the anonymous arithmetic printed at Bambcrg in 1483, had as their
common source a manuscript known as Algorismus Ratisponensis.
8 Nvmerandi doctrina authore Lodoico Baeza (Lvtctia, 1556), fol. 45.
9 C. Dibvadii in arithmeticam irralionalivm Evclidis (Arnhemii, 1605).
10 Paolo Casati, Fabrica et Vao Del Composso di Proportione (Bologna, 1685)
[Privilege, 1662], p. 33, 39, 43, 63, 125.
COMMON FRACTIONS 311
usually omitted in the writings of Marin Mersenne1 of 1644 and
1647. It is frequently but not usually omitted by Tobias Beutel.2
In the middle of a fourteenth-century manuscript3 one finds the
>— I HH
notation 3 5 for f, 4 7 for -4- A Latin manuscript,4 Paris 7377 A,
which is a translation from the Arabic of Abu Kamil, contains the
fractional line, as in J, but -J-J is a continued fraction and stands for j
plus ^r> whereas $ \ as well as -jj| represent simply -fa. Similarly,
Leonardo of Pisa,5 who drew extensively from the Arabic of Abu
Kamil, lets -85Jj- stand for -£f, there being a difference in the order of
reading. Leonardo read from right to left, as did the Arabs, while
authors of Latin manuscripts of about the fourteenth century read
as we do from left to right. In the case of a mixed number, like 3J,
Leonardo and the Arabs placed the integer to the right of the fraction.
274. Special symbols for simple fractions of frequent occurrence
are found. The Ahmes papyrus has special signs for J and | (§22);
there existed a hieratic symbol for j (§ 18). Diophantus employed
special signs for | and | (§ 104). A notation to indicate one-half,
almost identical with one sometimes used during the Middle Ages in
connection with Roman numerals, is found in the fifteenth century
with the Arabic numerals. Says Cappelli: "I remark that for the des-
ignation of one-half there was used also in connection with the Arabic
numerals, in the XV. century, a line between two points, as 4 -~ for
4J, or a small cross to the right of the number in place of an exponent,
as 4*, presumably a degeneration of 1/1, for in that century this form
was used also, as 7 1/1 for 7 J. Toward the close of the XV. century
one finds also often the modern form |."6 The Roman designation of
certain unit fractions are set forth in § 58. The peculiar designations
employed in the Austrian cask measures are found in § 89. In a fif-
teenth-century manuscript we find: "Whan pou hayst write pat, for
pat pat leues, write such a merke as is here w vpon his hede, pe quych
1 Marin Mersenne, Cogitata Phy&ico-mathemalica (Paris, 1644), "Phaenomena
ballistica"; Novarvm observationvm Physico-mathematicarvmy Tomvs III (Paris,
1647), p. 194 ff.
2 Tobias Beutel, Geometrische Gallerie (Leipzig, 1690), p. 222, 224, 236, 239,
240, 242, 243, 246.
3 Bibliothcca mathemdtica (3d ser.), Vol. VII, p. 308-9.
«L. C. Karpinski in ibid., Vol. XII (1911-12), p. 53, 54.
5 Leonardo of Pisa, Liber abbaci (ed. B. Boncompagni, 1857), p. 447. Note-
worthy here is the use of e to designate the absence of a number.
0 A. Cappelli, Lexicon Abbreviaturarum (Leipzig, 1901), p. L.
312 A HISTORY OF MATHEMATICAL NOTATIONS
merke schal betoken halfe of pe odde pat was take away";1 for ex-
ample, half of 241 is 120^. In a mathematical roll written apparently
in the south of England at the time of Recorde, or earlier, the char-
acter ~ stands for one-half, a dot • for one-fourth, and ~ for three-
fourths.2 In some English archives3 of the sixteenth and seventeenth
centuries one finds one-half written in the form ~j . In the earliest
arithmetic printed in America, the Arle para aprendar todo el menor del
arithmetica of Pedro Paz (Mexico, 1623), the symbol JL is used for | a
few times in the early part of the book. This symbol is taken from the
Arithmetica practica of the noted Spanish writer, Juan Perez de Moya,
1562 (14th ed., 1784, p. 13), who uses JQ. and also ° for ^ or media.
This may be a convenient place to refer to the origin of the sign
% for "per cent/' which has been traced from the study of manuscripts
by D. E. Smith.4 He says that in an Italian manuscript an "unknown
writer of about 1425 uses a symbol which, by natural stages, developed
into our present %. Instead of writing ' per 100', 'P 100' or T
cento/ as had commonly been done before him, he wrote 'Per-2'
o o
for 'I? 8/ just as the Italians wrote 1, 2, ... and 1°, 2°, ... for primo,
secundo, etc. In the manuscripts which I have examined the evolution
is easily traced, the o-* becoming -JJ- about 1650, the original meaning
having even then been lost. Of late the 'per' has been dropped;
leaving only {j or %." By analogy to %, which is now made up of two
zeros, there has been introduced the sign %0, having as many zeros
as 1,000 and signifying per milled Cantor represents the fraction
(100+/>)/100 "by the sign 1, Op, not to be justified mathematically
but in practice extremely convenient."
275. The solidus* — The ordinary mode of writing fractions r is
typographically objectionable as requiring three terraces of type. An
effort to remove this objection was the introduction of the solidus, as
in a/6, where all three fractional parts occur in the regular line of type.
It was recommended by De Morgan in his article on "The Calculus
1 R. Steele, The Earliest Arithmetics in English (London, 1922), p. 17, 19. The
p in "pou," "pat," etc., appears to be our modern th.
2 D. E. Smith in American Mathematical Monthly, Vol. XXIX (1922), p. 63.
3 Antiquaries Journal, Vol. VI (London, 1926), p. 272.
4 D. E. Smith, Rara arithmetica (1898), p. 439, 440.
5 Moritz Cantor, Politische Arithmetik (2. Aufl.; Leipzig, 1903), p. 4.
8 The word "solidus" in the time of the Roman emperors meant a gold coin
(a "solid" piece of money) ; the sign / comes from the old form of the initial letter s,
namely, f, just as £ is the initial of libra ("pound"), and d of denarius ("penny").
COMMON FRACTIONS 313
of Functions/' published in the Encyclopaedia Metropolitana (1845).
But practically that notation occurs earlier in Spanish America. In
the Gazetas de Mexico (1784), page 1, Manuel Antonio Valdes used a
curved line resembling the sign of integration, thus 1/4, 3/4; Henri
Cambuston1 brought out in 1843, at Monterey, California, a small
arithmetic employing a curved line in writing fractions. The straight
solidus is employed, in 1852, by the Spaniard Antonio Serra Y Oli-
veres.2 In England, De Morgan's suggestion was adopted by Stokes3
in 1880. Cayley wrote Stokes, "I think the 'solidus' looks very well
indeed . . . . ; it would give you a strong claim to be President of a
Society for the Prevention of Cruelty to Printers." The solidus is
used frequently by Stolz and Gmeiner.4
While De Morgan recommended the solidus in 1843, he used a: b
in his subsequent works, and as Glaisher remarks, "answers the pur-
pose completely and it is free from the objection to -f- viz., that the
pen must be twice removed from the paper in the course of writing
it."5 The colon was used frequently by Leibniz in writing fractions
(§ 543, 552) and sometimes also by Karsten,6 as in 1:3 = J; the -f-
was used sometimes by Cayley.
G. Peano adopted the notation b/a whenever it seemed con-
venient.7
Alexander Macfarlane8 adds that Stokes wished the solidus to take
the place of the horizontal bar, and accordingly proposed that the
terms immediately preceding and following be welded into one, the
welding action to be arrested by a period. For example, rn? — n2/
was to mean (w2 — n2)/(w2+w2), and a/bed to mean , ,, but a/bc-d
to mean ,- d. "This solidus notation for algebraic expressions oc-
oc
1 Henri Cambuston, Definition de las principals opcraciones de arismetica
(1843), p. 26.
2 Antonio Serra Y Oliveres, Manuel de la Tipografia Espanola (Madrid, 1852),
p. 71.
3 G. G. Stokes, Math, and Phys. Papers, Vol. I (Cambridge, 1880), p, vii.
See also J. Larmor, Memoirs and Scient. Corr. ofG. G. Stokes, Vol. I (1907), p. 397.
4 O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (2d cd.; Leipzig, 1911),
p. 81.
6 J. W. L. Glaisher, Messenger of Mathematics, Vol. II (1873), p. 109.
6 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, Vol. I (Greifswald,
1767), p. 50, 51, 55.
7 G. Peano, Lezioni di analisi infinitesimale, Vol. I (Torino, 1893), p. 2.
8 Alexander Macfarlane, Lectures on Ten British Physicists (New York, 1919),
p. 100, 101.
314 A HISTORY OF MATHEMATICAL NOTATIONS
curring in the text has since been used in the Encyclopaedia Britannica,
in Wiedemann's^nnafen and quite generally in mathematical litera-
ture." It was recommended in 1915 by the Council of the London
Mathematical Society to be used in the current text.
"The use of small fractions in the rnidst of letterpress," says
Bryan,1 "is often open to the objection that such fractions are difficult
to read, and, moreover, very often do not come out clearly in printing.
It is especially difficult to distinguish % from \ For this reason
it would be better to confine the use of these fractions to such common
forms as J, ^, f , |, and to use the nptation 18/22 for other fractions."
SIGNS OP DECIMAL FRACTIONS
276. Stevin's notation. — The invention of decimal fractions is
usually ascribed to the Belgian Simon Stevin, in his La Disme, pub-
lished in 1585 (§ 162). But at an earlier date several other writers
came so close to this invention, and at a later date other writers ad-
vanced the same ideas, more or less independently, that rival candi-
dates for the honor of invention were bound to be advanced. The
La Disme of Stevin marked a full grasp of the nature and importance
of decimal fractions, but labored under the burden of a clumsy nota-
tion. The work did not produce any immediate effect. It was trans-
lated into English by R. Norton2 in 1608, who slightly modified the
notation by replacing the circles by round parentheses. The frac-
tion .3759 is given by Norton in the form 3(1)7(2)5(3)9(4).
277. Among writers who adopted Stevin's decimal notation is
Wilhelm von Kalcheim3 who writes 693 @ for our 6.93. He applies it
also to mark the decimal subdivisions of linear measure: "Die Zeichen
sind diese: (o) ist ein ganzes oder eine ruthe: ® ist ein erstes / prime
oder schuh: @ ist ein zweites / secunde oder Zoll: ® ein drittes /
korn oder gran: @ ist ein viertes stipflin oder minuten: und so
forthan." Before this J. H. Beyer writes4 8 798 for 8.00798; also
1 G. H. Bryan, Mathematical Gazette, Vol. VIII (1917), p. 220.
2 Disme: the Art of Tenths, or Decimall Arithmetike, .... invented by the excel-
lent mathematician, Simon Stevin. Published in English with some additions by
Robert Norton, Gent. (London, 1608). See also A. de Morgan in Companion to
the British Almanac (1851), p. 11.
3 Zusammenfassung etlicher geomelrischen Aufgaben. .... Durch Wilhelra von
Kalcheim, genant Lohausen Obristen (Bremen, 1629), p. 117.
4 Johann Hartmann Beyer, Logistica decimalis, das ist die Kunstrechnung mil
den zehntheiligen Briichen (Frankfurt a/M., 1603). We have not seen Beyer's
DECIMAL FRACTIONS 315
viii 0 i ii iii iv v vi i ii iil iv v vi
14.3761 for 14.00003761, 123.4.5.9.8.7.2. or 123.4.5.9.8.7.2
or 123. 459. 872 for 123.459872, 643 for 0.0643.
That Stevin's notation was not readily abandoned for a simpler
one is evident from Ozanam's use1 of a slight modification of it as
(1) (2) (3) (4) (0) fl) (2)
late as 1691, in passages like "TWftr 6g. d 6 Q Q 7," and 3 9 8 for
our 3.98.
278. Other notations used before 1617. — Early notations which one
might be tempted to look upon as decimal notations appear in works
whose authors had no real comprehension of decimal fractions and
their importance. Thus Regiomontanus,2 in dividing 85869387 by
60000, marks off the last four digits in the dividend and then divides
by 6 as follows:
8586|9387
1431
In the same way, Pietro Borgi3 in 1484 uses the stroke in dividing
123456 by 300, thus
"per 300
1 2 3 4 | 5 6
4 1 1
411JU-"
Francesco Pellos (Pellizzati) in 1492, in an arithmetic published at
Turin, used a point and came near the invention of decimal fractions.4
Christoff Rudolff5 in his Coss of 1525 divides 652 by 10. His
words are: "Zu exempel / ich teile 652 durch 10. stet also 65/2. ist
65 der quocient vnnd 2 das iibrig. Kompt aber ein Zal durch 100 zu
teilen / schneid ab die ersten zwo figuren / durch 1000 die erstcn drey /
also weiter fur yede o ein figur." ("For example, I divide 652 by 10.
It gives 65/2; thus, 65 is the quotient and 2 the remainder. If a
number is to be divided by 100, cut off the first two figures, if by
book ; our information is drawn from J. Tropfke, Geschichte der Elementar-Mathc-
matik, Vol. I (2d ed.; Berlin and Leipzig, 1921), p. 143 ;S. Giinther, Geschichtc der
Mathcmatik, Vol. I (Leipzig, 1908), p. 342.
1 J. Ozanam, V 'Usage du Compas de Proportion (a La Haye, 1691), p. 203, 211.
2 AbhantUungen zur Geschichtc der Mathematik, Vol. XII (1902), p. 202, 225.
3 See G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. X (1909-10),
p. 240.
4 D. E. Smith, Rara arithmetica (1898), p. 50, 52.
6 Quoted by J. Tropfke, op. cit., Vol. I (2d ed., 1921), p. 140.
316 A HISTORY OF MATHEMATICAL NOTATIONS
1,000 the first three, and so on for each 0 a figure.") This rule for
division by 10,000, etc., is given also by P. Apian1 in 1527.
In the Exempel Biichlin (Vienna, 1530), Rudolff performs a
multiplication involving what we now would interpret as being deci-
mal fractions.2 Rudolff computes the values 375 (l+iiU)n for n=l,
2, ,10. For n = l he writes 393 | 75, which really denotes 393.75;
for ft = 3 he writes 434 | 109375. The computation for n = 4 is as fol-
lows:
434|109375
21 70546875
455|81484375
Here Rudolff uses the vertical stroke as we use the comma and, in
passing, uses decimals without appreciating the importance and'
generality of his procedure.
F. Vieta fully comprehends decimal fractions and speaks of the
advantages which they afford;3 he approaches close to the modern
notations, for, after having used (p. 15) for the fractional part
smaller type than for the integral part, he separated the decimal from
the integral part by a vertical stroke (p. 64, 65); from the vertical
stroke to the actual comma there is no great change.
In 1592 Thomas Masterson made a close approach to decimal frac-
tions by using a vertical bar as separatrix when dividing £337652643
by a million and reducing the result to shillings and pence. He wrote :4
facit
/. 337
s. 1 3
d.
6 5 2 6 4 3 "
052860
634320
John Kepler in his Oesterreichisches Wein-Visier-Buchlein (Lintz,
MDCXVI), reprinted in Kepler's Opera omnia (ed. Ch. Frisch),
Volume V (1864), page 547, says: "Furs ander, weil ich kurtze
Zahlen brauche, derohalben es offt Briiche geben wirdt, so mercke,
dass alle Ziffer, welche nach dem Zeichen (() folgen, die gehoren zu
1 P. Apian, Kauffmannsz Rechnung (Ingolstadt, 1527), fol. cttjr0. Taken from
J. Tropfke, op. tit., Vol. I (2d ed., 1921), p. 141.
2 See D. E. Smith, "Invention of the Decimal Fraction," Teachers College
Bulletin (New York, 1910-11), p. 18; G. Enestrom, Bibliotheca mathematica (3d
ser.), Vol. X (1909-10), p. 243.
3 F. Vieta, Universalium inspcclionum, p. 7; Appendix to the Canon mathe-
maticus (1st ed.; Paris, 1579). We copy this reference from the Encyclopedic des
scienc. m.ath., Tome I, Vol. I (1904), p. 53, n. 180.
4 A. de Morgan, Companion to the British Almanac (1851), p. 8.
DECIMAL FRACTIONS 317
dem Bruch, als der Zehlcr, der Nenner darzu wird nicht gesetzt, ist
aber allczcit cine runde Zehnerzahl von so vil Nullen, als vil Ziffcr
nach dem Zeichen kommen. Wann kcin Zeichen nicht ist, das ist
eine gantze Zahl ohne Bruch, vnd wann also alle Ziffern nach dem
Zeichen gehen, da hebcn sie bissweilen an von einer Nullen. Disc
Art der Bruch-rechnung ist von Jost Biirgcn zu der sinusrechnung
erdacht, vnd ist darzu gut, dass ich den Bruch abkiirtzen kan, wa er
vnnotig lang werden wil, ohne sonderen Schaden der vberigen Zahlen;
kan ihne auch etwa auff Erhaischung der Notdurfft crlengern. Item
lesset sich also die gantze Zahl vnd der Bruch mit einander durch
alle species Arithrneticae handlcn wie nur eine Zahl. Als wann ich
rechne 365 Gulden mit 6 per cento, wievil bringt es dess Jars Inter-
esse? dass stehet nun also :
3(65
6 mal
facit21(90
vnd bringt 21 Gulden vnd 90 hundertheil, oder 9 zchentheil, das ist
54 kr."
Joost Burgi1 wrote 1414 for 141.4 and 001414 for 0.01414; on the
0 o
title-page of his Progress-Tabulen (Prag, 1620) he wrote 230270022 for
our 230270.022. This small circle is referred to often in his Grundlicher
Unterrichty first published in 1856.2
279. Did Pitiscus use the decimal point? — If Bartholomaeus
Pitiscus of Heidelberg made use of the decimal point, he was probably
the first to do so. Recent writers3 on the history of mathematics are
1 See R. Wolf, Vicrtdj. Nalurf. Gcs. (Ziirich), Vol. XXXIII (1888), p. 226.
2 Gruncrt's Archiv der Mathcmatik und Physik, Vol. XXVI (1856), p. 316-34.
3 A. von Braunmuhl, Geschichte der Trigonometric, Vol. I (Leipzig, 1900), p. 225.
M. Cantor, Vorlcsungen iiber Geschichte der Mathematik, Vol. II (2d ed.;
Leipzig, 1913), p. 604, 619.
G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. VI (Leipzig, 1905),
p. 108, 109.
J. W. L. Glaisher in Napier Tercentenary Memorial Volume (London, 1913),
p. 77.
N. L. W. A. Gravclaar in Nieuw Archiefvoor Wiskunde (2d ser.; Amsterdam),
Vol. IV (1900), p. 73.
S. Giinther, Geschichte der Mathematik, 1. Toil (Leipzig, 1908), p. 342.
L. C. Karpinski in Science (2d ser.), Vol. XLV (New York, 1917), p. 663-65.
D. E. Smith in Teachers College Bulletin, Department of Mathematics (New
York, 1910-11), p. 19.
J. Tropfke, Geschichte der Elementar-Mathematik, Vol. I (2d ed. ; Leipzig, 1921),
p. 143.
318 A HISTORY OF MATHEMATICAL NOTATIONS
divided on the question as to whether or not Pitiscus used the decimal
point, the majority of them stating that he did use it. This disagree-
ment arises from the fact that some writers, apparently not having
access to the 1608 or 1612 edition of the Trigonometric,1 of Pitiscus,
reason from insufficient data drawn from indirect sources, while
others fail to carry conviction by stating their conclusions without
citing the underlying data.
Two queries are involved in this discussion: (1) Did Pitiscus
employ decimal fractions in his writings? (2) If he did employ them,
did he use the dot as the separatrix between units and tenths?
Did Pitiscus employ decimal fractions? As we have seen, the need
of considering -this question arises from the fact that some early
writers used a symbol of separation which we could interpret as
separating units from tenths, but which they themselves did not so
interpret. For instance,2 Christoff Rudolff in his Coss of 1525 divides
652 by 10, "stet also 65|2. ist 65 der quocientvnnd 2dasiibrig." The
figure 2 looks like two-tenths, but in RudohTs mind it is only a re-
mainder. With him the vertical bar served to separate the 65 from
this remainder; it was not a decimal separatrix, and he did not have
the full concept of decimal fractions. Pitiscus, on the other hand,
did have this concept, as we proceed to show. In computing the
chord of an arc of 30° (the circle having 107 for its radius), Pitiscus
makes the statement (p. 44): "All these chords are less than the
radius and as it were certain parts of the radius, which parts are com-
monly written iVsWoVtr- But much more brief and necessary for the
work, is this writing of it .05176381. For those numbers are alto-
gether of the same value, as these two numbers 09. and -fa are." In
the original Latin the last part reads as follows: " . . . . quae partes
vulgo sic scriberentur iVoVoWo • Sed multo compendiosior et ad
calculum accommodatior est ista scriptio .05176381. Omnino autem
idem isti numeri valent, sicut hi duo numeri 09. et VV idem valent."
One has here two decimals. The first is written .05176381. The
dot on the left is not separating units from tenths ; it is only a rhetorical
mark. The second decimal fraction he writes 09., and he omits the
dot on the left. The zero plays here the role of decimal separatrix.
1 1 have used the edition of 1612 which bears the following title: Bartholamci \
Pitisci Grunbergensis \ Silesij \ Trigonometriae \ Sioe. De dimensione Triangulor
[urn] Libri Qvinqve. Jtem \ Problematvm variorv. [m] nempe \ Geodaeticorum, \ Alti-
metricorum, \ Geographicorum, \ Gnom&nicorum, et \ Astronomicorum: \ Libri
Decem. \ Editio Tertia. \ Cui recens accessit Pro \ blematum Arckhiteclonicarum
Liber \ unus \ Franeofurti. \ Typis Nicolai Hofmanni: \ Surnptibus lonae Rosae]
M.DCXIL
2 Quoted from J. Tropfke, op. tit., Vol. I (1921), p. 140.
DECIMAL FRACTIONS 319
The dots appearing here are simply the punctuation marks written
after (sometimes also before) a number which appears in the running
text of most medieval manuscripts and many early printed books on
mathematics. For example, Clavius1 wrote in 1606: "Deinde quia
minor est \. quam $. erit per propos .8. minutarium libri 9. Euclid,
minor proportio 4. ad 7. quam 3. ad 5."
Pitiscus makes extensive use of decimal fractions. In the first
five books of his Trigonometria the decimal fractions are not preceded
by integral values. The fractional numerals are preceded by a zero;
thus on page 44 he writes 02679492 (our 0.2679492) and finds its
square root which he writes 05176381 (our 0.5176381). Given an arc
and its chord, he finds (p. 54) the chord of one-third that arc. This
leads to the equation (in modern symbols) 3o?—x3 = . 5176381, the
radius being unity. In the solution of this equation by approximation
he obtains successively 01, 017, 0174 .... and finally 01743114. In
computing, he squares and cubes each of these numbers. Of 017, the
square is given as 00289, the cube as 0004913. This proves that
Pitiscus understood operations with decimals. In squaring 017 ap-
pears the following:
"001.7
2 7
1 89
002 89.4"
What role do these dots play? If we put a = -1V> & — vJk> then
(a+6)2 = «2+(2a+6)6; 001 -a2, 027=(2a+6), 00189= (2a+&)6,
00289= (a+fr).2 The dot in 001.7 serves simply as a separator be-
tween the 001 and the digit 7, found in the second step of the approxi-
mation. Similarly, in 00289.4, the dot separates 00289 and the digit 4,
found in the third step of the approximation. It is clear that the dots
used by Pitiscus in the foreging approximation are not decimal
points.
The part of Pitiscus' Trigonometria (1612) which bears the title
"Problematvm variorvm .... libri vndecim" begins a new pagina-
tion. Decimal fractions are used extensively, but integral parts
appear and a vertical bar is used as decimal separatrix, as (p. 12)
where he says, "pro .... 13|00024. assumo 13. fractione scilicet
10 oo oa neglecta." ("For 13.00024 I assume 13, the fraction, namely,
fbo.Voo being neglected/') Here again he displays his understanding
of decimals, and he uses the dot for other purposes than a decimal
separatrix. The writer has carefully examined every appearance of
1 Christophori Clavius .... Geometria practica (Mogvntiac, 160G), p. 343-
320 A HISTORY OF MATHEMATICAL NOTATIONS
dots in the processes of arithmetical calculation, but has failed to
find the dot used as a decimal separatrix. There are in the Pitiscus
of 1612 three notations for decimal fractions, the three exhibited in
0522 (our .522), 5 1 269 (our 5.269), and the form (p. 9) of common
fractions, 121-jVoV In one case (p. 11) there occurs the tautological
notation 29|-,Vjr (our 29.95).
280. But it has been affirmed that Pitiscus used the decimal point
in his trigonometric Table. Indeed, the dot does appear in the
Table of 1612 hundreds of times. Is it used as a decimal point? Let
us quote from Pitiscus (p. 34) : "Therefore the radius for the making
of these Tables is to be taken so much the more, that there may be
no error in so many of the figures towards the left hand, as you will
have placed in the Tables: And as for the superfluous numbers they
are to be cut off from the right hand toward the left, after the ending
of the calculation. So did Regiomontanus, when he would calculate
the tables of sines to the radius of 6000000; he took the radius
60000000000. and after the computation was ended, he cut off from
every sine so found, from the right hand toward the left four figures, so
Rhaeticus when he would calculate a table of sines to the radius of
10000000000 took for the radius 1000000000000000 and after the
calculation was done, he cut off from every sine found from the right
hand toward the left five figures: But I, to find out the numbers in the
beginning of the Table, took the radius of 100000 00000 00000 00000
00000. But in the Canon itself have taken the radius divers numbers
for necessity sake: As hereafter in his place shall be declared."
On page 83 Pitiscus states that the radius assumed is unity fol-
lowed by 5, 7, 8, 9, 10, 11, or 12 ciphers, according to need. In solving
problems he takes, on page 134, the radius 107 and writes sin 61°46' =
8810284 (the number in the table is 88102.838); on page 7 ("Probl.
var.") he takes the radius 105 and writes sin 41°10' = 65825 (the num-
ber in the Table is 66825.16). Many examples are worked, but in no
operation are the trigonometric values taken from the Table written
down as decimal fractions. In further illustration we copy the fol-
lowing numerical values from the Table of 1612 (which contains sines,
tangents, and secants) :
" sin 2" = 97 sec 3" = 100000 . 00001 . 06
sin 3" = 1 . 45 sec 2°30' = 100095 . 2685 .
tan 3" = 1 . 45 sec 3°30' = 100186 . 869
sin 89°59'59" = 99999 . 99999 . 88
tan 89°59'59" = 20626480624 .
sin 30°31' = 50778 . 90 sec 30°31' = 116079 . 10*'
DECIMAL FRACTIONS 321
To explain all these numbers the radius must be taken 1012. The
100000.00001.06 is an integer. The dot on the right is placed be-
tween tens and hundreds. The dot on the left is placed between
millions and tens of millions.
When a number in the Table contains two dots, the left one is
always between millions and tens of millions. The right-hand dot is be-
tween tens and hundreds, except in the case of the secants of angles be-
tween 0°19' and 2°31/ and in the case of sines of angles between 87°59'
and 89°40'; in these cases the right-hand dot is placed (probably
through a printer's error) between hundreds and thousands (see sec.
2°30'). The tangent of 89°59'59" (given above) is really 20626480624-
0000000, when the radius is 1012. All the figures below ten millions are
omitted from the Table in this and similar cases of large functional
values.
If a sine or tangent has one dot in the Table and the secant for
the same angle has two dots, then the one dot for the sine or tangent
lies between millions and tens of millions (see sin 3", sec 3").
If both the sine and secant of an angle have only one dot in the
Table and r= 1012, that dot lies between millions and tens of millions
(see sin 30°31' and sec 30°31'). If the sine or tangent of an angle has
no dots whatever (like sin 2"), then the figures are located immedi-
ately below the place for tens of millions. For all angles above 2°30'
and below 88° the numbers in the Table contain each one and only
one dot. If that dot were looked upon as a decimal point, correct re-
sults could be secured by the use of that part of the Table. It would
imply that the radius is always to be taken 105. But this interpreta-
tion is invalid for any one of the following reasons: (1) Pitiscus does
not always take the r = 105 (in his early examples he takes r= 107), and
he explicitly says that the radius may be taken 105, 107, 108, 109, 1010,
1011, or 1012, to suit the degrees of accuracy demanded in the solution.
(2) In the numerous illustrative solutions of problems the numbers
taken from the Table are always in integral form. (3) The two dots
appearing in some numbers in the Table could not both be decimal
points. (4) The numbers in the Table containing no dots could not
be integers.
The dots were inserted to facilitate the selection of the trigono-
metric values for any given radius. For r= 105, only the figures lying
to the left of the dot between millions and tens of millions were copied.
For r=10l°, the figures to the left of the dot between tens and hun-
dreds were chosen, zeroes being supplied in cases like sin 30°31',
where there was only one dot, so as to yield sin 30°31' = 5077890000.
322 A HISTORY OF MATHEMATICAL NOTATIONS
For r = 107, the figures for 105 and the two following figures were
copied from the Table, yielding, for example, sin 30°31' = 5077890.
Similarly for other cases.
In a Table1 which Pitiscus brought out in 1613 one finds the sine
of 2°52/30" given as 5015.71617.47294, thus indicating a different
place assignment of the dots from that of 1612. In our modern tables
the natural sine of 2°52/30" is given as .05015. This is in harmony
with the statement of Pitiscus on the title-page that the Tables are
computed "ad radium 1.00000.00000.00000." The observation to be
stressed is that these numbers in the Table of Pitiscus (1613) are not
decimal fractions, but integers.
Our conclusions, therefore, are that Pitiscus made extended use
of decimal fractions, but that the honor of introducing the dot as the
separatrix between units and tenths must be assigned to others.
J. Ginsburg has made a discovery of the occurrence of the dot in
the position of a decimal separatrix, which he courteously permits to
be noted here previous to the publication of his own account of it.
He has found the dot in Clavius' Astrolabe, published in Rome in
1593, where it occurs in a table of sines and in the explanation of
that table (p. 228). The table gives sin 16°12' = 2789911 and sin
16°13' = 2792704. Clavius places in a separate column 46.5 as a cor-
rection to be made for every second of arc between 16°12' and 16°13'.
He obtained this 46.5 by finding the difference 2793 "between the
two sines 2789911.2792704," and dividing that difference by 60. He
identifies 46.5 as signifying 46t*V This dot separates units and tenths.
In his works, Clavius uses the dot regularly to separate any two suc-
cessive numbers. The very sentence which contains 46.5 contains also
the integers "2789911.2792704." The question arises, did Clavius in
that sentence use both dots as general separators of two pairs of
numbers, of which one pair happened to be the integers 46 and the
five-tenths, or did Clavius consciously use the dot in 46.5 in a more
restricted sense as a decimal separatrix? His use of the plural "duo
hi numeri 46,5" goes rather against the latter interpretation. If a
more general and more complete statement can be found in Clavius,
these doubts may be removed. In his Algebra of 1608, Clavius writes
all decimal fractions in the form of common fractions. Nevertheless,
Clavius unquestionably deserves a place in the history of the intro-
duction of the dot as a decimal separatrix.
More explicit in statement was John Napier who, in his Rabdologia
1 B. Pitiscus, Thesavrvs mathematicvs, sive Canon sinwn (Francofurti, 1613),
p. 19.
DECIMAL FRACTIONS 323
of 1617, recommended the use of a "period or comma" and uses the
comma in his division. Napier's Construdio (first printed in 1619) was
written before 1617 (the year of his death). In section 5 he says:
" Whatever is written after the period is a fraction," and he actually
uses the period. In the Leyden edition of the Construdio (1620) one
finds (p. 6) "25.803. idem quod 25M\."
281. The point occurs in E. Wright's 1616 edition of Napier's
Description but no evidence has been advanced, thus far, to show that
the sign was intended as a separator of units and tenths, and not as a
more general separator as in Pitiscus.
282. The decimal comma and point of Napier. — That John Napier
in his Rabdologia of 1617 introduced the comma and point as sepa-
rators of units and tenths, and demonstrated that the comma was
intended to be used in this manner by performing a division, and
properly placing the comma in the quotient, is admitted by all his-
torians. But there are still historians inclined to the belief that he was
not the first to use the point or comma as a separatrix between units
and tenths. We copy from Napier the following: "Since there is the
same facility in working with these fractions as with whole numbers,
you will be able after completing the ordinary division, and adding a
period or comma, as in the margin, to add to the dividend or to the
remainder one cypher to obtain
6 4
1 3 6
3 1 6
1 1 8,0 0 0
1 4 1
402
429
8 6 1 0 9 4,0 0 0(1 9 9 3,2 7 3
432
3888
3888
1296
864
3024
.1296
tenths, two for hundredths, three for thousandths, or more after-
wards as required: And with these you will be able to proceed with
the working as above. For instance, in the preceding example, here
repeated, to which we have added three cyphers, the quotient will
324 A HISTORY OF MATHEMATICAL NOTATIONS
become 1993,273, which signifies 1993 units and 273 thou-
sandth parts or WoV"1
Napier gives in the Rabdologia only three examples in which
decimals occur, and even here he uses in the text the sexagesimal ex-
ponents for the decimals in the statement of the results.2 Thus he
/ // /// ////
writes 1994.9160 as 1994,9 1 6 0 ; in the edition brought out at
Leyden in 1626, the circles used by S. Stevin in his notation of deci-
mals are used in place of Napier's sexagesimal exponents.
Before 1617, Napier used the decimal point in his Constructio,
where he explains the notation in sections 4, 5, and 47, but the Con-
structio was not published until 1619, as already stated above. In
section 5 he says: "Whatever is written after the period is a fraction,"
and he actually uses the period. But in the passage we quoted from
Rabdologia he speaks of a "period or comma" and actually uses a com-
ma in his illustration. Thus, Napier vacillated between the period
and the comma; mathematicians have been vacillating in this matter
ever since.
In the 1620 edition3 of the Constructio, brought out in Leyden,
one reads: "Vt 10000000.04, valet idem, quod 1 0000000 rU- Item
25.803. idem quod 25 ^V Item 9999998.0005021, idem valet quod
9999998iTH-Uinr- & sic de caeteris."
283. Seventeenth-century notations after 1617. — The dot or comma
attained no ascendancy over other notations during the seventeenth
century.
In 1623 John Johnson (the survaighour)* published an Arithmatick
which stresses decimal fractions and modifies the notation of Stevin
by omitting the circles. Thus, £ 3. 2 2 9 1 6 is written
1. 2. 3. 4. 5.
£3 22916,
while later in the text there occurs the symbolism 31 | 2500 and
54)2625, and also the more cautious "358149411 fifths" for our
358.49411.
1 John Napier, Rabdologia (Edinburgh, 1617), Book I, chap. iv. This passage
is copied by W. R. Macdonald, in his translation of John Napier's Constructio
(Edinburgh, 1889), p. 89.
2J. W. L. Glashier, "Logarithms and Computation," Napier Tercentenary
Memorial Volume (ed. Cargill Gilston Knott; London, 1915), p. 78.
3 Mirifwi logarithmorvm Canonis Constructio .... authore & Inventore loanne
Nepero, Barone Merchistonii, etc, (Scoto. Lvgdvni, M.DC.XX.), p. 6.
4 From A. de Morgan in Companion to the British Almanac (1851), p. 12.
DECIMAL FRACTIONS 325
Henry Briggs1 drew a horizontal line under the numerals in the
decimal part which appeared in smaller type and in an elevated posi-
tion; Briggs wrote 59J^ for our 5.9321. But in his Tables of 1624 he
employs commas, not exclusively as a decimal separatrix, although
one of the commas used for separation falls in the right place between
units and tenths. He gives -0,22724,3780 as the logarithm of ff.
A. Girard2 in his Invention nouvelle of 1629 uses the comma on one
occasion ; he finds one root of a cubic equation to be IfVW and then
explains that the three roots expressed in decimals are 1,532 and 347
and —1,879. The 347 is .347; did Girard consider the comma un-
necessary when there was no integral part?
Burgi's and Kepler's notation is found again in a work which
appeared in Poland from the pen of Joach. Stegman;3 he writes
39(063. It occurs again in a geometry written by the Swiss Joh.
Ardiiser.4
William Oughtrcd adopted the sign 2|5 in his Clavis mathematicae
of 1631 and in his later publications.
In the second edition of Wingate's Arithmetic (1650; ed. John
Kersey) the decimal point is used, thus: .25, .0025.
In 1651 Robert Jager5 says that the common way of natural arith-
metic being tedious and prolix, God in his mercy directed him to what
he published; he writes upon decimals, in which 16|7249 is our
16.7249.
Richard Balam6 used the colon and wrote 3:04 for our 3.04. This
same symbolism was employed by Richard Rawly ns,7 of Great Yar-
mouth, in England, and by H. Meissner8 in Germany.
1 Henry Briggs, Arithmetica logarithmica (London, 1624), Lectori. S.
2 De Morgan, Companion to the British Almanac (1851), p. 12; Invention nou-
velle, fol. E2.
3 Joach. Stegman, Institutionum mathematicarum libri II (Rakow, 1630), Vol.
I, cap. xxiv, "De logistica decimali." We take this reference from J. Tropfke,
op. cit., Vol. I (2d ed., 1921), p. 144.
4 Joh. Ardttser, Geometriae theoricae et practicae XII libri (Zurich, 1646), fol.
306, 1SO&, 27()a.
5 Robert Jager, Artificial Arithmclick in Decimals (London, 1651). Our infor-
mation is drawn from A. de Morgan in Companion to the British Almanac (1851),
p. 13.
6 Rich. Balam, Algebra (London, 1653), p. 4.
7 Richard Rawlyns, Practical Arilhmetick (London, 1656), p. 262.
8H. Meissner, Geometria tyronica (1696[?]). This reference is taken from
J. Tropfke, op. cit.. Vol. I (2d ed!, 1921), p. 144.
326 A HISTORY OF MATHEMATICAL NOTATIONS
Sometimes one encounters a superposition of one notation upon
another, as if one notation alone might not be understood. Thus F. van
Schooten1 writes 58,5 © for 58.5, and 638,82 © for 638.82. Tobias
11 111 IV V
Beutel2 writes 645. JJfo. A. Tacquet3 sometimes writes 25.8 0079,
at other times omits the dot, or the Roman superscripts.
Samuel Foster4 of Gresham College, London, writes 31. k^;
he does not rely upon the dot alone, but adds the horizontal line
found in Briggs.
Johann Caramuel5 of Lobkowitz in Bohemia used two horizontal
parallel lines, like our sign of equality, as 22 = 3 for 22.3, also 92 =
123,345 for 92.123345. In a Parisian text by Jean Prestet6 272097792
is given for 272.097792; this mode of writing had been sometimes
used by Stevin about a century before Prestet, and in 1603 by Beyer.
William Molyneux7 of Dublin had three notations; he frequently
used the comma bent toward the right, as in 30t24. N. Mercator8 in
his Logarithmotechnia and Dechales9 in his course of mathematics
used the notation as in 12[345.
284. The great variety of forms for separatrix is commented on by
Samuel Jeake in 1696 as follows: "For distinguishing of the Decimal
Fraction from Integers, it may truly be said, Quot Homines, lot Sen-
tentiae; every one fancying severally. For some call the Tenth Parts,
the Primes; the Hundredth Parts, Seconds; the Thousandth Parts,
Thirds, etc. and mark them with Indices equivalent over their heads.
As to express 34 integers and iW<ftr Parts of an Unit, they do it thus,
/ // /// //// (1) (2) (3) (4)
34.1. 4. 2. 6. Or thus, 34.1. 4. 2. 6. Others thus, 34,1426""; or thus,
34,1426^4>. And some thus, 34.1 . 4 . 2 . 6 . setting the Decimal Parts
1 Francisci a Schooten, Exercitalionvm mathemaiicarum liber primus (Leyden,
1657), p. 33, 48, 49.
2 Tobias Beutel, Geometrischer Lust-Garten (Leipzig, 1690), p. 173.
3 Arithmeticae theoria et praxis, autore Andrea Tacqvet (2d ed.; Antwerp, 1665),
p. 181-88.
4 Samuel Foster, Miscellanies: or Mathematical Lvcvbrations (London, 1659),
p. 13.
6 Joannis Caramvels Mathesis Biceps. Vetus, et Nova (Companiae, 1670),
"Arithmetica," p. 191.
6 Jean Prestet, Nouveaux elcmens des mathematiques, Premier volume (Paris,
1689), p. 293.
7 William Molyneux, Treatise of Dioplricks (London, 1692), p. 165.
8 N. Mercator, Logarithmotechnia (1668), p. 19.
9 A. de Morgan, Companion to the British Almanac (1851), p. 13.
DECIMAL FRACTIONS 327
at little more than ordinary distance one from the other Others
distinguish the Integers from the Decimal Parts only by placing a
Coma before the Decimal Parts thus, 34,1426; a good way, and very
useful. Others draw a Line under the Decimals thus, 34 L4-3-6-, writing
them in smaller Figures than the Integers. And others, though they
use the Coma in the work for the best way of distinguishing them, yet
after the work is done, they use a Rectangular Line after the place of
the Units, called Separatrix, a separating Line, because it separates the
Decimal Parts from the Integers, thus 34 [1426. And sometimes the
Coma is inverted thus, 34' 1426, contrary to the true Coma, and set at
top. I sometimes use the one, and sometimes the other, as cometh to
hand." The author generally uses the comma. This detailed state-
ment from this seventeenth-century writer is remarkable for the
omission of the point as a decimal separatrix.
285. Eighteenth-century discard of clumsy notations. — The chaos in
notations for decimal fractions gradually gave way to a semblance of
order. The situation reduced itself to trials of strength between the
comma and the dot as separatrices. To be sure, one finds that over a
century after the introduction of the decimal point there were authors
who used besides the dot or comma the strokes or Roman numerals to
indicate primes, seconds, thirds, etc. Thus, Chelucci1 in 1738 writes
o i n in iv i iv ii v
5.8 6 4 2, also4.2 5 for 4.2005, 3.5 7for3.05007.
W. Whiston2 of Cambridge used the semicolon a few times, as in
0;9985, though ordinarily he preferred the comma. O. Gherli8 in
Modena, Italy, states that some use the sign 35 1 345, but he himself
uses the point. E. Wells4 in 1713 begins with 75.25, but later in his
arithmetic introduces Oughtred's J75. Joseph Raphson's transla-
tion into English of I. Newton's Universal Arithmetick (1728) ,5 con-
tains 732,[569 for our 732.569. L'Abb6 Deidier6 of Paris writes the
1 Paolirio Chelucci, Institutions analyticae .... auctore Paulino A. S.
Josepho Lucensi (Rome), p. 35, 37, 41, 283.
2 Isaac Newton, Arithmetica Vniversalis (Cambridge, 1707), edited by
G. W[histon], p. 34.
3 O. Gherli, Gli dementi .... delle mathematiche pure, Vol. I (Modena, 1770),
p. 60.
4 Edward Wells, Young gentleman's arithmetick (London, 1713), p. 59, 105, 157.
6 Universal Arithmetick, or Treatise of Arithmetical Composition and Resolu-
tion .... transl. by the late Mr. Joseph Ralphson, & revised and corrected by Mr.
Cunn (2d ed.; London, 1728), p. 2.
6 L'Abbe" Deidier, U Arithmetique des Gtometres, ou nouveaux eUmens de mathe'-
matiques (Paris, 1739), p. 413.
328 A HISTORY OF MATHEMATICAL NOTATIONS
decimal point and also the strokes for tenths, hundredths, etc. He
says: "Pour ajouter ensemble 32.6' 3" 4'" et 8.5' 4".3'"—
32 6 3 4111
854 3111
41 1 7 7m "
A somewhat unusual procedure is found in Sherwin's Tables1 of 1741,
where a number placed inside a parenthesis is used to designate the
number of zeroes that precede the first significant figure in a decimal ;
thus, (4) 2677 means .00002677.
In the eighteenth century, trials of strength between the comma
and the dot as the separatrix were complicated by the fact that Leib-
niz had proposed the dot as the symbol of multiplication, a proposal
which was championed by the German textbook writer Christian
Wolf and which met with favorable reception throughout the Conti-
nent. And yet Wolf2 himself in 1713 used the dot also as separatrix,
as "loco 5i ffVinF scribimus 5.0047." As a symbol for multiplication the
dot was seldom used in England during the eighteenth century,
Oughtred's X being generally preferred. For this reason, the dot as
a separatrix enjoyed an advantage in England during the eighteenth
century which it did not enjoy on the Continent. Of fifteen British
books of that period, which we chose at random, nine used the dot and
six the comma. In the nineteenth century hardly any British authors
employed the comma as separatrix.
In Germany, France, and Spain the comma, during the eighteenth
century, had the lead over the dot, as a separatrix. During that
century the most determined continental stand in favor of the dot
was made in Belgium3 and Italy.4 But in recent years the comma has
finally won out in both countries.
1 H. Sherwin, Mathematical Tables (3d ed. ; rev. William Gardiner, London,
1741), p. 48.
2 Christian Wolf, Elementa matheseos universae, Tomus I (Halle, 1713), p. 77.
3 De'sire' Andre4, Des Notations MatMmatiques (Paris, 1909), p. 19, 20.
4 Among eighteenth-century writers in Italy using the dot are Paulino A. S.
Josepho Lucensi who in his Institutiones analyticae (Rome, 1738) uses it in con-
nection with an older symbolism, "3.05007"; G. M. della Torre, Istituzioni arim-
metiche (Padua, 1768) ; Odoardo Gherli, Elementi delle matematiche pure, Modena,
Tomo I (1770); Peter Ferroni, Magnitudinum exponentialium logarithmorum et
trigonometriae sublimis theoria (Florence, 1782); F. A. Tortorella, Arithmetica
degVidioti (Naples, 1794).
DECIMAL FRACTIONS 329
286. Nineteenth century: different positions for dot and comma. —
In the nineteenth century the dot became, in England, the favorite
separatrix symbol. When the brilliant but erratic Randolph Churchill
critically spoke of the "damned little dots," he paid scant respect to
what was dear to British mathematicians. In that century the dot
came to serve in England in a double capacity, as the decimal symbol
and as a symbol for multiplication.
Nor did these two dots introduce confusion, because (if we may
use a situation suggested by Shakespeare) the symbols were placed in
Romeo and Juliet positions, the Juliet dot stood on high, above
Romeo's reach, her joy reduced to a decimal over his departure, while
Romeo below had his griefs multiplied and was "a thousand times the
worse" for want of her light. Thus, 25 means 2^, while 2.5 equals
10. It is difficult to bring about a general agreement of this kind,
but it was achieved in Great Britain in the course of a little over half
a century. Charles Hutton1 said in 1795: "I place the point near the
upper part of the figures, as was done also by Newton, a method which
prevents the separatrix from being confounded with mere marks of
punctuation." In the Latin edition2 of Newton's Arithmetica uni-
versalis (1707) one finds, "Sic numerus 732'|569. denotat septingentas
triginta duas imitates, .... qui et sic 732,|569, vel sic 732*569. vel
ctiam sic 732j569, nunnunquam scribitur .... 57104*2083 ....
0'064." The use of the comma prevails; it is usually placed high, but
not always. In Horsely's and Castillon's editions of Newton's Arith-
metica universalis (1799) one finds in a few places the decimal nota-
tion 3572; it is here not the point but the comma that is placed on
high. Probably as early as the time of Hutton the expression "deci-
mal point" had come to be the synonym for "separatrix" and was
used even when the symbol was not a point. In most places in Hors-^
ley's and Castillon's editions of Newton's works, the comma 2,5 is
used, and only in rare instances the point 2.5. The sign 2 ' 5 was used
in England by H. Clarke3 as early as 1777, and by William Dickson4
in 1800. After the time of Hutton the 2 • 5 symbolism was adopted by
Peter Barlow (1814) and James Mitchell (1823) in their mathematical
dictionaries. Augustus de Morgan states in his Arithmetic: "The
1 Ch. Hutton, Mathematical and Philosophical Dictionary (London, 1795),
art. "Decimal Fractions."
2 1. Newton, Arithmetica universalis (ed. W. Whiston; Cambridge, 1707), p. 2.
Sec also p. 15, 16.
3 H. Clarke, Rationale of Circulating Numbers (London, 1777).
4 W. Dickson in Philosophical Transactions, Vol. VIII (London, 1800), p. 231.
330 A HISTORY OF MATHEMATICAL NOTATIONS
student is recommended always to write the decimal point in a line
with the top of the figures, or in the middle, as is done here, and never
at the bottom. The reason is that it is usual in the higher branches
of mathematics to use a point placed between two numbers or letters
which are multiplied together."1 A similar statement is made in 1852
by T. P. Kirkman.2 Finally, the use of this notation in Todhunter's
texts secured its general adoption in Great Britain.
The extension of the usefulness of the comma or point by assign-
ing it different vertical positions was made in the arithmetic of Sir
Jonas Moore3 who used an elevated and inverted comma, 116*64.
This notation never became popular, yet has maintained itself to
the present time. Daniel Adams,4 in New Hampshire, used it, also
Juan de Dios Salazar5 in Peru, Don Gabriel Ciscar6 of Mexico, A. de
la Rosa Toro7 of Lima in Peru, and Federico Villareal8 of Lima.
The elevated and inverted comma occurs in many, but not all, the
articles using decimal fractions in the Enciclopedia-vniversal ilvstrada
Evropeo-Americana (Barcelona, 1924).
Somewhat wider distribution was enjoyed by the elevated but not
inverted comma, as in 2'5. Attention has already been called to the
occurrence of this symbolism, a few times, in Horsley's edition of
Newton's Arithmetica universalis. It appeared also in W. Winston's
edition of the same work in 1707 (p. 15). Juan de Dios Salazar of
Peru, who used the elevated inverted comma, also uses this. It is
Spain and the Spanish-American countries which lead in the use of
this notation. De La-Rosa Toro, who used the inverted comma, also
used this. The 2'5 is found in Luis Monsante9 of Lima; in Maximo
1 A. de Morgan, Elements of Arithmetic (4th cd.; London, 1840), p. 72.
2 T. P. Kirkman, First Mnemonical Lessons in Geometry, Algebra and Trigo-
nometry (London, 1852), p. 5.
3 Moore's Arithmetick: In Four Books (3d ed.; London, 1688), p. 369, 370,
465.
4 Daniel Adams, Arithmetic (Keene, N.H., 1827), p. 132.
5 Juan de Dios Salazar, Lecciones de Aritmetica, Teniente del Cosmografo
major de esta Republica del Peru (Arequipa, 1827), p. 5, 74, 126, 131. This book
has three diiTerent notations: 2,5; 2'5; 2*5.
6 Don Gabriel Ciscar, Curso de esludios elementales de Marina (Mexico, 1825).
7 Agustin de La-Rosa Toro, Aritmetica Teorico-Proxtica (tercera ed. ; Lima,
1872), p. 157.
8 D. Federico Villareal, Calculo Binomial (P. I. Lima [Peru], 1898), p. 416.
9 Luis Monsante, Lecciones de Aritmetica Demostrada (7th ed. ; Lima, 1872),
p. 89.
DECIMAL FRACTIONS 331
Vazquez1 of Lima; in Manuel Torres Torija2 of Mexico; in D. J.
Cortazar3 of Madrid. And yet, the Spanish-speaking countries did
not enjoy the monopoly of this symbolism. One finds the decimal
comma placed in an elevated position, 2'5, by Louis Bertrand4 of
Geneva, Switzerland.
Other writers use an inverted wedge-shaped comma,5 in a lower
position, thus: 2^5. In Scandinavia and Denmark the dot and the
comma have had a very close race, the comma being now in the lead.
The practice is also widely prevalent, in those countries, of printing
the decimal part of a number in smaller type than the integral part.6
Thus one frequently finds there the notations 2,5 and 2.5. To sum up,
in books printed within thirty-five years we have found the decimal
notations7 2-5, 2*5, 2,5, 2'5, 2*5, 2A5, 2,5, 2.6.
287. The earliest arithmetic printed on the American continent
which described decimal fractions came from the pen of Greenwood,8
professor at Harvard College. He gives as the mark of separation "a
Comma, a Period, or the like," but actually uses a comma. The arith-
metic of "George Fisher" (Mrs. Slack), brought out in England, and
also her The American Instructor (Philadelphia, 1748) contain both
the comma and the period. Dil worth's The Schoolmaster's Assistant,
an English book republished in America (Philadelphia, 1733), used
the period. In the United States the decimal point9 has always had the
1 Maximo Vazquez, Aritmetica practica (septiema ed.; Lima, 1875), p. 57.
2 Manuel Torres Torija, Nociones de Algebra Superior y elemcntos fundamen-
tales de cdlculo differencial e Integral (Mexico, 1894), p. 137.
3 D. J. Cortazdr, Tratado de Aritmetica (42d ed.; Madrid, 1904).
4 L. Bertrand, Developpment nouveaux de la partie elemenlaire des mathe-
matiques, Vol. I (Geneva, 1778), p. 7.
5 As in A. F. Vallin, Aritmetica para los ninos (41st ed.; Madrid, 1889), p. 66.
6 Gustaf Haglund, Samlying of Ofningsexempel till Ldrabok i Algebra, Fjerde
Upplagan (Stockholm, 1884), p. 19; Ofversigt af Kongl. Vetenskaps-Akademiens
Forhandlingar, Vol. LIX (1902; Stockholm, 1902, 1903), p. 183, 329; Oversigl over
del Kongelige Danske Videnskabernes Selskabs, Fordhandlinger (1915; Kobenhavn,
1915), p. 33, 35, 481, 493, 545.
7 An unusual use of the elevated comma is found in F. G. Gausz's Funfstellige
vollstdndige Logar. u. Trig. Tafeln (Halle a. S., 1906), p. 125; a table of squares of
numbers proceeds from AT = 0'00 to N = lO'OO. If the square of 03 is wanted, take
the form 6'3; its square is 39'6900. Hence 632 = 3969.
8 Isaac Greenwood, Arithmctick Vulgar and Decimal (Boston, 1729), p. 49.
See facsimile of a page showing decimal notation in L. C. Karpinski, History of
Arithmetic (Chicago, New York, 1925), p. 134.
9 Of interest is Chauncey Lee's explanation in his American Accomptant
(Lasingburgh, 1797), p. 54, that, in writing denominate numbers, he separates
332 A HISTORY OF MATHEMATICAL NOTATIONS
lead over the comma, but during the latter part of the eighteenth
and the first half of the nineteenth century the comma in the position
of 2,5 was used quite extensively. During 1825-50 it was the influence
of French texts which favored the comma. We have seen that Daniel
Adams used 2'5 in 1827, but in 1807 he1 had employed the ordinary
25,17 and ,375. Since about 1850 the dot has been used almost ex-
clusively. Several times the English elevated dot was used in books
printed in the United States. The notation 2*5 is found in Thomas
Sarjeant's Arithmetic? in F. Nichols' Trigonometry? in American
editions of Hutton's Course of Mathematics that appeared in the in-
terval 1812-31, in Samuel Webber's Mathematics? in William Griev's
Mechanics Calculator, from the fifth Glasgow edition (Philadelphia,
1842), in The Mathematical Diary of R. Adrain5 about 1825, in
Thomas Sherwin's Common School Algebra (Boston, 1867; 1st ed.,
1845), in George R. Perkins' Practical Arithmetic (New York, 1852).
Sherwin writes: "To distinguish the sign of Multiplication from the
period used as a decimal point, the latter is elevated by inverting the
type, while the former is larger and placed down even with the lower
extremities of the figures or letters between which it stands." In
1881 George Bruce Halsted6 placed the decimal point halfway up and
the multiplication point low.
It is difficult to assign definitely the reason why the notation 2*5
failed of general adoption in the United States. Perhaps it was due
to mere chance. Men of influence, such as Benjamin Peirce, Elias
Loomis, Charles Davies, and Edward Olncy, did not happen to be-
come interested in this detail. America had no one of the influence
of De Morgan and Todhunter in England, to force the issue in favor
of 2*5. As a result, 2. 5 had for a while in America a double meaning,
namely, 2 5/10 and 2 times 5. As long as the dot was seldom used to
the denominations "in a vulgar table" by two commas, but "in a decimal table"
by the decimal point; he writes £ 175,, 15,, 9, and 1.41.
Daniel Adams, Scholar's Arithmetic (4th ed.; Keene, N.H., 1807).
2 Thomas Sarjeant, Elementary Principles of Arithmetic (Philadelphia, 1788),
p. 80.
3F. Nichols, Plane and Spherical Trigonometry (Philadelphia, 1811), p. 33.
4 Samuel Webber, Mathematics, Vol. I (Cambridge, 1801; also 1808, 2d cd.),
p. 227.
6 R. Adrain, The Mathematical Diary, No. 5, p. 101.
6 George Bruce Halsted, Elementary Treatise on Mensuration (Boston, 1881).
DECIMAL FRACTIONS 333
express multiplication, no great inconvenience resulted, but about 1880
the need of a distinction arose. The decimal notation was at that
time thoroughly established in this country, as 2.5, and the dot for
multiplication was elevated to a central position. Thus with us 2-5
means 2 times 5.
Comparing our present practice with the British the situation is
this: We write the decimal point low, they write it high; we place the
multiplication dot halfway up, they place it low. Occasionally one
finds the dot placed high to mark multiplication also in German books,
as, for example, in Friedrich Meyer1 who writes 2 ' 3 = 6.
288. It is a notable circumstance that at the present time the
modern British decimal notation is also the notation in use in Austria
where one finds the decimal point placed high, but the custom does not
seem to prevail through any influence emanating from England. In
the eighteenth century P. Mako2 everywhere used the comma, as in
3,784. F. S. Mozhnik3 in 1839 uses the comma for decimal fractions,
as in 3,1344, and writes the product "2 . 3..n." The Sitzungsberichte
der philosophisch-historischen Classe d. K. Akademie der Wissenschaften,
Erster Band (Wien, 1848), contains decimal fractions in many articles
and tables, but always with the low dot or low comma as decimal
scparatrix ; the low dot is used also for multiplication, as in "1.2.3. . .r."
But the latter part of the nineteenth century brought a change.
The decimal point is placed high, as in 1*63, by I. Lernoch4 of Lembcrg.
N. Fialkowski of Vienna in 1863 uses the elevated dot5 and also in
1892.6 The same practice is followed by A. Steinhauser of Vienna,7
by Johann Spielmann8 and Richard Supplantschitsch,9 and by Karl
1 Friedrich Meyer, Driller Cursus der Planimelrie (Halle a/8., 1885), p. 5.
2 P. Mako e S.I., De . . . . aequalionvm resolvlionibvs libri dvo (Vienna, 1770),
p. 135; Compendiaria Malhcseos Institvtio Pavlvs Mako e S.I. in Coll. Keg.
Thcres Prof. Math, et Phys. Experim. (editio tertia; Vienna, 1771).
3 Franz Scraphin Mozhnik, Theorie der numcrischen Glcichungen (Wien, 1839),
p. 27, 33.
4 Ignaz Lemoch, Lchrbuch der praklischen Geomelrie, 2. Theil, 2. Aufl. (Wien,
1857), p. 163.
6 Nikolaus Fialkowski, Das Decimalrcchnen mil Rangziffern (Wien, 1863), p. 2.
6 N. Fialkowski, Praktische Geomclrie (Wien, 1892), p. 48.
7 Anton Steinhauser, Lehrbuch der Mathematik. Algebra (Wien, 1875), p. Ill,
138.
8 Johann Spielmann, Motniks Lehrbuch der Geometric (Wien, 1910), p. 66.
9 Richard Supplantschitsch, Malhemalisches Unlerrichlswerk, Lehrbuch der
Geomelrie (Wien, 1910), p. 91.
334 A HISTORY OF MATHEMATICAL NOTATIONS
Rosenberg.1 Karl Zahradnfcek2 writes 0-35679.1'0765.1'9223.0'3358,
where the lower dots signify multiplication and the upper dots are
decimal points. In the same way K. Wolletz3 writes ( — 0'0462).
0-0056.
An isolated instance of the use of the elevated dot as decimal
separatrix in Italy is found in G. Peano.4
In France the comma placed low is the ordinary decimal separa-
trix in mathematical texts. But the dot and also the comma are used
in marking off digits of large numbers into periods. Thus, in a political
and literary journal of Paris (1908)6 one finds "2,251,000 drachmes,"
"Fr. 2.638.370 75," the francs and centimes being separated by a
vacant place. One finds also "601,659 francs 05" for Fr. 601659. 05.
It does not seem customary to separate the francs from centimes by a
comma or dot.
That no general agreement in the notation for decimal fractions
exists at the present time is evident from the publication of the In-
ternational Mathematical Congress in Strasbourg (1920), where deci-
mals are expressed by commas6 as in 2,5 and also by dots7 as in 2.5.
in that volume a dot, placed at the lower border of a line, is used also
to indicate multiplication.8
The opinion of an American committee of mathematicians is
expressed in the following: "Owing to the frequent use of the letter x,
it is preferable to use the dot (a raised period) for multiplication in
the few cases in which any symbol is necessary. For example, in a
case like 1-2-3 . . . . (x— l)-z, the center dot is preferable to the
symbol X; but in cases like 2a(x — a) no symbol is necessary. The
committee recognizes that the period (as in a. 6) is more nearly
international than the center dot (as in a -6); but inasmuch as the
period will continue to be used in this country as a decimal point,
1 Karl Rosenberg, Lehrbuch der Physik (Wien, 1913), p. 125.
2 Karl Zahradnf£ek, Mocniks Lehrbuch der Arithmetik und Algebra (Wien,
1911), p. 141.
3K. Wolletz, Arithmetik und Algebra (Wien, 1917), p. 163.
4 Giuseppe Peano, Risoluzione graduate delle equazioni numeriche (Torino,
1919), p. 8. Reprint from Atti delta r. Accad. delle Scienze di Torino, Vol. LIV
(1918-19).
6 Les Annales, Vol. XXVI, No. 1309 (1908), p. 22, 94.
6 Comptes rendus du congres international des mathematiques (Strasbourg,
22-30 Septembre 1920; Toulouse, 1921), p. 253, 543, 575, 581.
7 Op. cit., p. 251. •
< Op. tit., p. 1.53,252,545.
POWERS 335
it is likely to cause confusion, to elementary pupils at least, to attempt
to use it as a symbol for multiplication. "l
289. Signs for repeating decimals. — In the case of repeating deci-
mals, perhaps the earliest writer to use a special notation for their
designation was John Marsh,2 who, "to avoid the Trouble for the
future of writing down the Given Repetend or Circulate, whether
Single or Compound, more than once," distinguishes each "by placing
a Period over the first Figure, or over the first and last Figures of the
given Repetend." Likewise, John Robertson3 wrote 0,3 for 0,33 . . . . ,
0,23 for 0,2323 , 0,785 for 0,785785 H. Clarke4 adopted
/ / /
the signs .6 for .666 . . . . , .642 for .642642 A choice favoring
the dot is shown by Nicolas Pike5 who writes, 379, and by Robert
Pott6 and James Pryde7 who write *3, '45, '34567. A return to ac-
cents is seen in the Dictionary of Davies and Peck8 who place accents
over the first, or over the first and last figure, of the repetend, thus:
.'2, .'5723', 2.418'.
SIGNS OF POWERS
290. General remarks.— An ancient symbol for squaring a number
occurs in a hieratic Egyptian papyrus of the late Middle Empire,
now in the Museum of Fine Arts in Moscow.9 In the part containing
the computation of the volume of a frustrated pyramid of square
base there occurs a hieratic term, containing a pair of walking legs
J\ and signifying "make in going," that is, squaring the number. The
Diophantine notation for powers is explained in § 101, the Hindu
notation in §§ 106, 110, 112, the Arabic in § 116, that of Michael
Psellus in § 117. The additive principle in marking powers is referred
1 The Reorganization of Mathematics in Secondary Schools, by the National Com-
mittee on Mathematical Requirements, under the auspices of the Mathematical
Association of America (1923), p. 81.
2 John Marsh, Decimal Arithmetic Made Perfect (London, 1742), p. 5.
3 John Robertson, Philosophical Transactions (London, 1768), No. 32, p. 207-
13. See Tropfke, op. cit., Vol. I (1921), p. 147.
4 H. Clarke, The Rationale of Circulating Numbers (London, 1777), p. 15, 16.
6 Nicolas Pike, A New and Complete System of Arithmetic (Newbury-port,
1788), p. 323.
8 Robert Pott, Elementary Arithmetic, etc. (Cambridge, 1876), Sec. X, p. 8.
7 James Pryde, Algebra Theoretical and Practical (Edinburgh, 1852), p. 278.
8 C. Davies and W. G. Peck, Mathematical Dictionary (1855), art. ' 'Circulating
Decimal."
9 See Ancient Egypt (1917), p. 100-102.
336 A HISTORY OF MATHEMATICAL NOTATIONS
to in §§ 101, 111, 112, 124. The multiplicative principle in marking
powers is elucidated in §§ 101, 111, 116, 135, 142.
Before proceeding further, it seems desirable to direct attention
to certain Arabic words used in algebra and their translations into
Latin. There arose a curious discrepancy in the choice of the princi-
pal unknown quantity; should it be what we call x, or should it be #2?
al-Khowarizmi and the older Arabs looked upon a;2 as the principal
unknown, and called it mal ("assets," "sum of money").1 This view-
point may have come to them from India. Accordingly, x (the Arabic
jidr, "plant-root," "basis," "lowest part") must be the square root of
mal and is found from the equation to which the problem gives rise.
By squaring x the sum of money could be ascertained.
Al-Khowarizmi also had a general term for the unknown, shai
("thing"); it was interpreted broadly and could stand for either mal
or jidr (z2 or x). Later, John of Seville, Gerard of Cremona, Leonardo
of Pisa, translated the Arabic jidr into the Latin radix, our x; the
Arabic shai into res. John of Seville says in his arithmetic:2 "Quaeri-
tur ergo, quae res cum. X. radicibus suis idem decies acccpta radice
sua efficiat 39." ("It is asked, therefore, what thing together with 10
of its roots or what is the same, ten times the root obtained from it,
yields 39.") This statement yields the equation £2+10x = 39. Later
shai was also translated as causa } a word which Leonardo of Pisa
used occasionally for the designation of a second unknown quantity.
The Latin res was translated into the Italian word cosa, and from
that evolved the German word coss and the English adjective "cossic."
We have seen that the abbreviations of the words cosa arid cubus,
viz., co. and cu., came to be used as algebraic symbols. The words
numerus, dragma, denarius, which were often used in connection with
a given absolute number, experienced contractions sometimes em-
ployed as symbols. Plato of Tivoli,3 in his translation from the Hebrew
of the Liber embadorum of 1145, used a new term, latus ("side"),
for the first power of the unknown, x, and the name embadum ("con-
tent") for the second power, x2. The term latus was found mainly
in early Latin writers drawing from Greek sources and was used later
by Ramus (§ 322), Vieta (§ 327), and others.
291. Double significance of "R" and "1." — There came to exist
considerable confusion on the meaning of terms and symbols, not only
1 J. Ruska, Sitzungxberichte Heidelberger Akad.} PhiL-hist. Klasse (1917),
Vol. II, p. 61 f.; J. Tropfke, op. ciL, Vol. II (2d ed., 1921), p. 106.
2Tropfke, op. cit., Vol. II (2d ed., 1921), p. 107.
8 M. Curtze, Bibliotheca mathematica (3d scr.), Vol. I (1900), p. 322, n. 1.
POWERS 337
because res (x) occasionally was used for x2, but more particularly
because both radix and latus had two distinct meanings, namely, x
and Vx. The determination whether x or Vx was meant in any par-
ticular case depended on certain niceties of designation which the
unwary was in danger of overlooking (§ 137).
The letter I (latus) was used by Ramus and Vieta for the designa-
tion of roots. In some rare instances it also represented the first power
of the unknown x. Thus, in Schoner's edition of Ramus1 51 meant 5x,
while 15 meant 1/5. Schoner marks the successive powers "/., q., c.,
bq., J., qc., 6.J., tq., cc." and named them latus, quadratus, cubus,
biquadratus, and so on. Ramus, in his Scholarvm mathematicorvm
libri unus et triginti (1569), uses the letter / only for square root, not
for x or in the designation of powers of x; but he uses (p. 253) the
words latus, quadratus, latus cubi for x, xz, a;3.
This double use of I is explained by another pupil of Ramus,
Bernardus Salignacus,2 by the statement that if a number precedes
the given sign it is the coefficient of the sign which stands for a power
of the unknown, but if the number comes immediately after the I
the root of that number is to be extracted. Accordingly, 2q, 3c, 51
stand respectively for 2x2, 3z3, 5x; on the other hand, 15, IcS, IbqW
stand respectively for 1/5, ^8, V 16. The double use of the capital L
is found in G. Gosselin (§§ 174, 175).
B. Pitiscus3 writes our 3z — z3 thus, 31!— Ic, and its square 9<? —
G.bq+lqc, while Willebrord Snellius4 writes our 5x — 5x3+a:5 in the
form 51 5c+l/3. W. Oughtred5 writes x5-15.r4+160x3-1250o;2+
6480s =170304782 in the form lqc-l5qq+lGQc-l25Qq+Q48Ql =
170304782.
Both /. and R. appear as characters designating the first power
1 Petri Rami Veromandui Philosophi .... arithmetica libri duo et geometriae
septem et viginti. Dudum quidem a Lazaro Schonero .... (Francofvrto ad moe-
nvm, MDCXXVII). P. 139 begins: "De Nvmeris figvratis Lazari Schoneri liber."
See p. 177.
2 Bernard 'i Salignaci Burdegalensis Algebrae libri duo (Francofurti, 1580).
See P. Trcutlein in Abhandl. zur Geschichte der Mathematik, Vol. II (1879),
p. 36.
3 Batholomaei Pitisd .... Trigonometriae editio tertia (Francofurti, 1612),
p. 60.
4 Willebrord Snellius, Doctrinae triangvlorvm cononicae liber qvatvor (Leyden,
1627), p. 37.
6 William Oughtred, Clavis mathematicae, under "De aequationum affectarum
resolutione in numeris" (1647 and later editions).
338 A HISTORY OF MATHEMATICAL NOTATIONS
in a work of J. J. Heinlin1 at Tubingen in 1679. He lets N stand for
unitas, numerus absolutus, if , /., R. for latus vel radix; z., q. for quad-
rains, zensus; ce, c for cubus; zz, qq, bq for biquadratics. But he utilizes
the three signs *f,lyR also for indicating roots. He speaks2 of "Latus
cubicum, vel Radix cubica, cujus nota est Lc. R.c. if. ce."
John Wallis3 in 1655 says "Est autem lateris I, numerus pyramida-
lis 13+W+21" and in 1685 writes4 "ll-2laa+a*: +bb," where the I
takes the place of the modern x and the colon is a sign of aggregation,
indicating that all three terms are divided by 62.
292. The use of I}. (Radix) to signify root and also power is seen
in Leonardo of Pisa (§ 122) and in Luca Pacioli (§§ 136, 137). The
sign /? was allowed to stand for the first power of the unknown x by
Peletier in his algebra, by K. Schott5 in 1661, who proceeds to let
Q. stand for x2, C. for x3, Biqq or qq. for x4, Ss. for x5, Cq. for x6, SsB.
for x7, Trig, or qqq for x8, Cc. for x9. One finds # in W. Leybourn's
publication of J. Billy's6 Algebra, where powers are designated by the
capital letters N, R, Q, QQ, 8, QC, 52, QQQ., and where x2 = 20-x is
written "lQ = 20-lfi."
Years later the use of R. for x and of !}. (an inverted capital letter
E, rounded) for x2 is given by Tobias Beutel7 who writes "21 £,
gleich 2100, 1 (?. gleigh 100, IR. gleich 10."
293. Facsimilis of symbols in manuscripts. — Some of the forms for
radical signs and for x, x2, x3, x4, and x5, as found in early German
manuscripts and in Widman's book, are tabulated by J. Tropfke, and
we reproduce his table in Figure 104.
In the Munich manuscript cosa is translated ding; the symbols in
Figure 104, C2a, seem to be modified d's. The symbols in C3 are signs
for res. The manuscripts C3b, C6, C7, C9, H6 bear on the evolution
of the German symbol for x. Paleographers incline to the view that it
is a modification of the Italian co, the o being highly disfigured. In
B are given the signs for dragma or numerus.
1 Joh. Jacobi Heirdini Synopsis mathematica universalis (3d ed. ; Tubingen,
1679), p. 66.
2 Ibid., p. 65. 3 John Wallis, Arithmetica infinitorum (Oxford, 1655), p. 144.
4 John Wallis, Treatise of Algebra (London, 1685), p. 227.
6 P. Gasparis Schotti .... Cursus mathematicus (Herbipoli [Wiirtzburg],
1661), p. 530.
8 Abridgement of the Precepts of Algebra. The Fourth Part. Written in French
by James Billy and now translated into English Published by Will. Ley-
bourn (London, 1678), p. 194.
7 Tobias Beutel, Geomelrische Galleri (Leipzig, 1690), p. 165.
POWERS
339
294. Two general plans for marking powers. — In the early develop-
ment of algebraic symbolism, no signs were used for the powers of given
numbers in an equation. As given numbers and coefficients were not
represented by letters in equations before the time of Vieta, but were
specifically given in numerals, their powers could be computed on the
spot and no symbolism for powers of such numbers was needed. It was
different with the unknown numbers, the determination of which con-
stituted the purpose of establishing an equation. In consequence,
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one finds the occurrence of symbolic representation of the unknown
and its powers during a period extending over a thousand years before
the introduction of the literal coefficient and its powers.
For the representation of the unknown there existed two general
plans. The first plan was to use some abbreviation of a name signify-
ing unknown quantity and to use also abbreviations of the names
signifying the square and the cube of the unknown. Often special
symbols were used also for the fifth and higher powers whose orders
were prime numbers. Other powers of the unknown, such as the
fourth, sixth, eighth powers, were represented by combinations of
those symbols. A good illustration is a symbolism of Luca Paciola,
in which co. (cosa) represented x, ce. (censo) x2, cu. (cube) re3, p.r.
(primo relato) x6; combinations of these yielded ce.ce. for x4, ce.cu. for
340 A HISTORY OF MATHEMATICAL NOTATIONS
x6, etc. We have seen these symbols also in Tartaglia and Cardan,
in the Portuguese Nuftez (§ 166), the Spanish Perez de Moya in 1652,
and Antich Rocha1 in 1564. We may add that outside of Italy
Pacioli's symbols enjoyed their greatest popularity in Spain. To be
sure, the German Marco Aurel wrote in 1552 a Spanish algebra (§ 165)
which contained the symbols of Rudolff, but it was Perez de Moya
and Antich Rocha who set the fashion, for the sixteenth century in
Spain; the Italian symbols commanded some attention there even
late in the eighteenth century, as is evident from the fourteenth un-
revised impression of Perez de Moya's text which appeared at
Madrid in 1784. The 1784 impression gives the symbols as shown in
Figure 105, and also the explanation, first given in 1562, that the
printing office does not have these symbols, for which reason the
ordinary letters of the alphabet will be used.2 Figure 105 is interesting,
for it purports to show the handwritten forms used by De Moya.
The symbols are not the German, but are probably derived from them.
In a later book, the Tratado de Mathematicas (Alcala, 1573), De Moya
gives on page 432 the German symbols for the powers of the unknown,
all except the first power, for which he gives the crude imitation Ze.
Antich Rocha, in his Arithmetica, folio 253, is partial to capita i
letters and gives the successive powers thus: N, Co, Ce, Cu, Cce, k,
CeCu, RR, Ccce, Ccu, etc. The same fondness for capitals is shown i i
his Mew for "more" (§320).
We digress further to state that the earliest mathematical work
published in America, the Sumario compendioso of Juan Diez Freyle3
1 Arithmelica por Antich Rocha de Gerona compuesta, y de varies Auctores
recopilada (Barcelona, 1564, also 1565).
2 Juan Perez de Moya, Aritmetica practica, y especulativa (14th ed.; Madrid,
1784), p. 263: "Por los diez caracte"res, que en el precedente capftulo se pusieron,
uso estos. Por el qual dicen numero n. por la cosa, co. por el censo, ce. por cubo,
cu. por censo, de censo, cce. por el primero relato, R. por el censo, y cubo, ce.cu.
por segundo relato, RR. por censo de censo de censo, cce. por cubo de cubo, ecu.
Esta figura r. quiere decir raiz quadrada, Esta figura rr. denota raiz quadrada de
raiz quadrada. Estas rrr. denota raiz cubica. De estos dos caract6res, p. m.
notards, que la p. quiere decir mas, y la m. menos, el uno es copulativo, el otro
disyuntivo, sirven para sumar, y restar cantidades diferentes, como adelante mejor
eriteriderjis. Quando despues de r. se pone u. denota raiz quadrada universal:
y asi rru. raiz de raiz quadrada universal: y de esta suerte rrru. raiz cubica uni-
versal. Esta figura ig. quiere decir igual. Esta q. denota cantidad, y asi qs. canti-
dades: estos caractc*res me ha parecido poner, porque no habia otros en la Impren-
ta; tu podras usar, q nan do hagas demandas, de los que se pusicron en el segundo
capitulo, porque son rnas breves, en lo dcmds todos son de una condicion."
3 Edition by D. E. Smith (Boston and London, 1921).
POWERS 341
(City of Mexico, 1556) gives six pages to algebra. It contains the
words cosa, zenso, or censo, but no abbreviations for them. The work
does not use the signs + or — , nor the p and m. It is almost purely
rhetorical.
The data which we have presented make it evident that in Perez de
Moya, Antich Rocha, and P. Nunez the symbols of Pacioli are used
and that the higher powers are indicated by the combinations of
symbols of the lower powers. This general principle underlies the no-
tations of Diophantus, the Hindus, the Arabs, and most of the Ger-
mans and Italians before the seventeenth century. For convenience
we shall call this the "Abbreviate Plan."
Cap. II. En el qualsepontn algunos car after es> quesirwnpor
cantidades proportionates.
En este capitulo se ponen algunos cara&eres , dando a cada
imo el nombre y valor que le conviene. Los quales son inveu-
tados por causa de brevedad; y es de saber, que no es de nece-
sidad, que estos, y no otros hayan de ser, porque cada uno pue-
de usar de lo que cjuisiere , e inventar mucho mas , procedien-
do con Ja proporcion que le pareciere. Los cara&eres son estos.
FIG. 105. — The written algebraic symbols for powers, as given in Perez de
Moya/s Arithmetica (Madrid, 1784), p. 260 (1st ed., 1562). The successive sym-
bols are called cosa es raiz, cen.so, cubo, censo de censo, primero relato, censo y cubo,
segimdo rdato, censo de censo de censo, cubo de cubo.
The second plan was not to use a symbol for the unknown quantity
itself, but to limit one's self in some way to simply indicating by a
numeral the power of the unknown quantity. As long as powers of
only one unknown quantity appeared in an equation, the writing of
the index of its power was sufficient. In marking the first, second,
third, etc., powers, only the numerals for "one," "two," "three,"
etc., were written down. A good illustration of this procedure is
Chuquet's 102 for lOz2, 101 for Wx, and 10° for 10. We shall call this
the "Index Plan." It was stressed by Chuquet, and passed through
several stages of development in Bombelli, Stevin, and Girard. Then,
after the introduction of special letters to designate one or more un-
known quantities, and the use of literal coefficients, this notation was
perfected by H£rigone and Hume; it finally culminated in the present-
day form in the writings of Descartes, Wallis, and Newton.
342 A HISTORY OF MATHEMATICAL NOTATIONS
295. Early symbolisms. — In elaborating the notations of powers
according to the "Abbreviate Plan" cited in § 294, one or the other
of two distinct principles was brought into play in combining the
symbols of the lower powers to mark the higher powers. One was the
additive principle of the Greeks in combining powers; the other was
the multiplicative principle of the Hindus. Diophantus expressed
the fifth power of the unknown by writing the symbols for x2 and
for y?y one following the other; the indices 2 and 3 were added. Now,
Bhaskara writes his symbols for x2 and y? in the same way, but lets
the two designate, not s5, but rr6; the indices 2 and 3 are multiplied.
This difference in designation prevailed through the Arabic period,
the later Middle Ages in Europe down into the seventeenth century.
It disappeared only when the notations of powers according to the
"Abbreviate Plan" passed into disuse. References to the early sym-
bolisms, mainly as exhibited in our accounts of individual authors,
are as follows:
ABBREVIATE PLAN
ADDITIVE PRINCIPLE
Diophantus, and his editors Xylander, Bachet, Fermat (§ 101)
al-Karkhi, eleventh century (§116)
Leonardo of Pisa (§ 122)
Anonymous Arab (§ 124)
Dresden Codex C. 80 (§ 305, Fig. 104)
M. Stifel, (1545), sum; sum; sum: x* (§ 154)
F. Vieta (1591), and in later publications (§ 177)
C. Glorioso, 1527 (§ 196)
W. Oughtred, 1631 (§ 182)
Samuel Foster, 1659 (§ 306)
MULTIPLICATIVE PRINCIPLE
Bhaskara, twelfth century (§ 110-12)
Arabic writers, except al-Karkhi (§ 116)
L. Pacioli, 1494, ce. cu. for z6 (§ 136)
H. Cardano, 1539, 1545 (§ 140)
N. Tartaglia, 1556-60 (§ 142)
Ch. Rudolff, 1525 (§ 148)
M. Stifel, 1544 (§ 151)
J. Scheubel, 1551, follows Stifel (§ 159)
A. Rocha, 1565, follows Pacioli (§ 294)
C. Clavius, 1608, follows Stifel (§ 161)
P. Nufiez, 1567, follows Pacioli and Cardan (§ 166)
R. Recorde, 1557, follows Stifel (§ 168)
POWERS 343
L. and T. Digges, 1579 (§ 170)
A. M. Visconti, 1581 (§ 145)
Th. Masterson, 1592 (§ 171)
J. Peletier, 1554 (§ 172)
G. Gosselin, 1577 (§ 174)
L. Schemer, 1627 (§291)
NEW NOTATIONS ADOPTED
Ghaligai and G. del Sodo, 1521 (§ 139)
M. Stifel, 1553, repeating factors (§ 156)
J. Buteon, 1559 (§ 173)
J. Scheubel, N, Ra, Pri, Se (§ 159)
Th. Harriot, repeating factors (§ 188)
Johann Geysius, repeating factors (§ 196, 305)
John Newton, 1654 (§305)
Nathaniel Torporley (§ 305)
Joseph Raphson, 1702 (§ 305)
Samuel Foster, 1659, use of lines (§ 300)
INDEX PLAN
Psellus, nomenclature without signs (§ 117)
Neophytos, scholia (§§ 87, 88)
Nicole Oresine, notation for fractional powers (§ 123)
N. Chuquet, 1484, 123 for 12r* (§ 131)
E. de la Roche, 1520 (§ 132)
R. Bombelli, 1572 (§ 144)
Grammateus, 1518, pri, se., ter. quart. (§ 147)
G. van der Hoecke, 1537, pri, se, 3a (§ 150)
S. Stevin (§ 162)
A. Girard, 1629 (§ 164)
L. & T. Digges, 1579 (§ 170, Fig. 76)
P. HSrigone, 1634 (§ 189)
J. Hume, 1635, 1636 (§ 190)
296. Notations applied only to an unknown quantity, the base being
omitted. — As early as the fourteenth century, Oresme had the ex-
ponential concept, but his notation stands in historical isolation and
does not constitute a part of the course of evolution of our modern
exponential symbolism. We have seen that the earliest important
steps toward the modern notation were taken by the Frenchman
Nicolas Chuquet, the Italian Rafael Bombelli, the Belgian Simon
Stevin, the Englishmen L. and T. Digges. Attention remains to be
called to a symbolism very similar to that of the Digges, which was
contrived by Pietro Antonio Cataldi of Bologna, in an algebra of 1610
344 A HISTORY OF MATHEMATICAL NOTATIONS
and a book on roots of 1613. Cataldi wrote the numeral exponents in
their natural upright position,1 and distinguished them by crossing
them out. His "5 3 via 8 if a 40 7" means 5z3 • 8#4=40o;7. His sign for
x is Z. He made only very limited use of this notation.
The drawback of Stevin's symbolism lay in the difficulty of writing
and printing numerals and fractions within the circle. Apparently as a
relief from this cumbrousness, we find that the Dutch writer, Adrianus
Romanus, in his Ideae Mathematicae pars prima (Antwerp, 1593),
uses in place of the circle two rounded parentheses and vinculums
above and below; thus, with him 1(45) stands for z45. He uses this
notation in writing his famous equation of the forty-fifth degree.
Franciscus van Schooten2 in his early publications and when he quotes
from Girard uses the notation of Stevin.
A notation more in line with Chuquet's was that of the Swiss
Joost Biirgi who, in a manuscript now kept in the library of the ob-
servatory at Pulkowa, used Roman numerals for exponents and wrote3
8+12-9+10+3+7-4 for 8z6+12z5-9z4+10^+3z2+7:r--4 .
In this notation Biirgi was followed by Nicolaus Reymers (1601) and
J, Kepler.4 Reymers5 used also the cossic symbols, but chose R in
place of 7£; occasionally he used a symbolism as in 25IIII+20II —
10III-8I for the modern 25z4+20r2- 10z3-8z. We see that Cataldi,
Romanus, Fr. van Schooten, Biirgi, Reymers, and Kepler belong in
the list of those who followed the "Index Plan."
297. Notations applied to any quantity, the base being designated. —
As long as literal coefficients were not used and numbers were not
generally represented by letters, the notations of Chuquet, Bombelli,
1 G. Wertheim, Zeitschr. f. Math. u. Physik, Vol. XLIV (1899), Hist.-Lit.
Abteilung, p. 48.
2 Francisci a Schooten, De Organica conicarum sectionum .... Tractatus
(Leyden, 1646), p. 96; Schooten, Renali Descartes Geometria (Frankfurt a./MM
1695), p. 359.
3 P. Treutlein in Abhandlungen zur Oeschichte der Mathemalik, Vol. II (Leipzig,
1879), p. 36, 104.
4 In his "De Figurarum regularium" in Opera omnia (ed. Ch, Frisch), Vol. V
(1864), p. 104, Kepler lets the radius AB of a circle be 1 and the side BC of a
regular inscribed heptagon be R. He says: "In hac proportione continuitatem
fingit, ut sicut est ABl ad BC IR, sic sit IR ad Iz, et 10 as 1 c£, et 1 c£ ad Izz, et
Izzad Iz <£ et sic perpetuo, quod nos commodius signabimus per apices six, 1, I1,
1", 1"', 1IV, F, ivi; IVH, etc."
6 N. Raimarus Ursus, Arithmetica analytica (Frankfurt a. O., 1601), Bl. C3v°.
See J. Tropfke, op. tit., Vol. II (2d ed., 1921), p. 122.
POWERS 345
Stevin, and others were quite adequate. There was no pressing need
of indicating the powers of a given number, say the cube of twelve;
they could be computed at once. Moreover, as only the unknown
quantity was raised to powers which could not be computed on the
spot, why should one go to the trouble of writing down the base?
Was it not sufficient to put down the exponent and omit the base?
v
Was it not easier to write 16 than IGx5? But when through the inno-
vations of Vieta and others, literal coefficients came to be employed,
and when several unknowns or variables came to be used as in ana-
lytic geometry, then the omission of the base became a serious defect
ii ii
in the symbolism. It will not do to write 15z2 — 16?/2 as 15—16. In
watching the coming changes in notation, the reader will bear this
problem in mind. Vieta/s own notation of 1591 was clumsy: D quadra-
turn or D. quad, stood for D2, D cubum for D8; A quadr. for z2, A repre-
senting the unknown number.
In this connection perhaps the first writer to be mentioned is
Luca Pacioli who in 1494 explained, as an alternative notation of
powers, the use of R as a base, but in place of the exponent he employs
an ordinal that is too large by unity (§ 136). Thus R. 30* stood for
a:29. Evidently Pacioli did not have a grasp of the exponential concept.
An important step was taken by Romanus1 who uses letters and
writes bases as well as the exponents in expressions like
A (4) +£(4) +4^1(3) in £+6A(2) in B(2)+±A in B(3)
which signifies
A similar suggestion came from the Frenchman, Pierre H6rigone, a
mathematician who had a passion for new notations. He wrote our a3
as a3, our 264 as 264, and our 2fea2 as 2ba2. The coefficient was placed
before the letter, the exponent after.
In 1636 James Hume2 brought out an edition of the algebra of
Vieta, in which he introduced a superior notation, writing down the
base and elevating the exponent to a position above the regular line
and a little to the right. The exponent was expressed in Roman
1 See H. Bosnians in Annales Societe sclent, de Bruxelles, Vol. XXX, Part II
(1906), p. 15.
2 James Hume, L'Algebre de Viete, d'une methode nouvelle claire et facile (Paris
1636). See (Euvres de Descartes (ed. Charles Adam et P. Tannery), Vol. V, p. 604,
506-12.
346 A HISTORY OF MATHEMATICAL NOTATIONS
numerals. Thus, he wrote AJii for A3. Except for the use of the Ro-
man numerals, one has here our modern notation. Thus, this Scots-
man, residing in Paris, had almost hit upon the exponential symbolism
which has become universal through the writings of Descartes.
298. Descartes9 notation of 1637. — Thus far had the notation ad-
vanced before Descartes published his Geometric (1637) (§ 191).
Hdrigone and Hume almost hit upon the scheme of Descartes. The
only difference was, in one case, the position of the exponent, and, in
the other, the exponent written in Roman numerals. Descartes ex-
pressed the exponent in Arabic numerals and assigned it an elevated
position. Where Hume would write 5aiv and H6rigone would write
5a4, Descartes wrote 5a4. From the standpoint of the printer, H6ri-
gone's notation was the simplest. But Descartes' elevated exponent
offered certain advantages in interpretation which the judgment of
subsequent centuries has sustained. Descartes used positive integral
exponents only.
299. Did Stampioen arrive at Descartes' notation independently? —
Was Descartes alone in adopting the notation 5a4 or did others hit
upon this particular form independently? In 1639 this special form
was suggested by a young Dutch writer, Johan Stampioen.1 He makes
no acknowledgment of indebtedness to Descartes. He makes it ap-
pear that he had been considering the two forms 3a and a3, and had
found the latter preferable.2 Evidently, the symbolism a3 was adopted
by Stampioen after the book had been written; in the body of his
book3 one finds aaa, bbbb, fffff, gggggg, but the exponential notation
above noted, as described in his passage following the Preface, is not
used. Stampioen uses the notation a4 in some but not all parts of a
controversial publication4 of 1640, on the solution of cubic equations,
and directed against Waessenaer, a personal friend of Descartes. In
view of the fact that Stampioen does not state the originators of any
of the notations which he uses, it is not improbable that his a3 was
taken from Descartes, even though Stampioen stands out as an
opponent of Descartes.5
1 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel (The Hague,
1639). See his statement following the Preface.
2 Stampioen's own words are: "aw* dit is a drievoudich in hem selfs gemen-
nichvuldicht. men soude oock daer voor konnen stellen 8a ofte better aV
8 J. Stampioen, op. tit., p. 343, 344, 348.
4 7. /. Stampioenii Wis-Konstigh ende Reden-Maetigh Bewys ('s Graven-
Hage, 1640), unpaged Introduction and p. 52-55.
6 (Euvres de Descartes, Vol. XII (1910), p. 32, 272-74.
POWERS 347
300. Notations used by Descartes before 1637. — Descartes' indebt-
edness to his predecessors for the exponential notation has been
noted. The new features in Descartes' notation, 5a3, 6a64, were in-
deed very slight. What notations did Descartes himself employ before
1637?
In his Opuscules de 1619-1621 he regularly uses German symbols
as they are found in the algebra of Clavius; Descartes writes1
"36~32-67ea65u.lt/'
which means 36 — 3#2— 6# = :c3. These Opuscules were printed by
Toucher de Careil (Paris, 1859-60), but this printed edition contains
corruptions in notation, due to the want of proper type. Thus the
numeral 4 is made to stand for the German symbol J^ ; the small letter
7 is made to stand for the radical sign j/« The various deviations
from the regular forms of the symbols are set forth in the standard
edition of Descartes' works. Elsewhere (§ 264) we call attention
that Descartes2 in a letter of 1640 used the Recordian sign of equality
and the symbols N and C of Xy lander, in the expression "1C— QN =
40." Writing to Mersenne, on May 3, 1638, Descartes3 employed the
notation of Vieta, "Aq+Bq+A in B bis" for our a2+&2+2a6. In a
posthumous document,4 of which the date of composition is not
known, Descartes used the sign of equality found in his Geomitrie
of 1637, and P. H£rigone's notation for powers of given letters, as
b3x for b*x, a3z for a?z. Probably this document was written before
1637. Descartes5 used once also the notation of Dounot (or Deidier, or
Bar-le-Duc, as he signs himself in his books) in writing the equation
lC-$Q+13Neq. 1/288-15, but Descartes translates it into t/3-9y2+
13!/- 121/2+ 15 ^0 0.
301. Use of Herigone's notation after 1 637.— After 1637 there was
during the seventeenth century still very great diversity in the ex-
ponential notation. H6rigone's symbolism found favor with some
writers. It occurs in Florimond Debeaune's letter6 of September 25,
1638, to Mersenne in terms like 2?/4, 2/3, 2Z2 for 2^, i/3, and 2P, re-
1 Ibid., Vol. X (1908), p. 249-51. See also E. de Jonquieres in Bibliotheca
mathematica (2d ser.), Vol. IV (1890), p. 52, also G. Enestrom, Bibliotheca mathe-
matwa (3d ser.), Vol. VI (1905), p. 406.
2 (Euvres de Descartes, Vol. Ill (1899), p. 190.
*Ibid., Vol. II (1898), p. 125; also Vol. XII, p. 279.
4 Ibid., Vol. X (1908), p. 299.
5 Ibid., Vol. XII, p. 278. 9 Ibid., Vol. V (1903), p. 516.
348 A HISTORY OF MATHEMATICAL NOTATIONS
spectively. G. Schott1 gives it along with older notations. Pietro
Mengoli2 uses it in expressions like a4+4a3r+6a2r2+4ar3+r4 for our
a4+4asr+6o2r2+4ar3+r4. The Italian Cardinal Michelangelo Ricci3
writes "ACZ in CB3" for 1~C2.(JB3. In a letter4 addressed to Ozanam
one finds 64+c4^a4 for 64+c4 = a4. Chr. Huygens5 in a letter of June
8, 1684, wrote a3+aab for a8+a26. In the same year an article by
John Craig6 in the Philosophical Transactions contains a3y+aA for
a*y+a*t but a note to the "Benevole Lector" appears at the end apolo-
gizing for this notation. Dechales7 used in 1674 and again in 1690
(along with older notations) the form A4:+4A3B+QA2B2+4:AB3 +
54. A Swedish author, Andreas Spole,8 who in 1664-66 sojourned
in Paris, wrote in 1692 an arithmetic containing expressions 3a3+
3a2-2a-2 for 3a3+3a2-2a-2. Joseph Moxon9 lets "A-B.(2)"
stand for our (A~B)\ also "A-B.(3)" for our (A-B)*. With the
eighteenth century this notation disappeared.
302. Later use of Hume's notation of 1636. — Hume's notation of
1636 was followed in 1638 by Jean de Beaugrand10 who in an anony-
mous letter to Mersenne criticized Descartes and states that the equa-
tion zlv+4sln- 19s11 -106s- 120 has the roots +5, -2, -3, -4.
Beaugrand also refers to Vieta and used vowels for the unknowns, as
in "A'"+3AAB+ADP esgale a ZS8." Again Beaugrand writes
"E'"<=> -13E-12" for ^-13^-12, where the o apparently desig-
nates the omission of the second term, as does •)(- with Descartes.
303. Other exponential notations suggested after 1637. — At the time
of Descartes and the century following several other exponential
notations were suggested which seem odd to us and which serve to
1 G. Schott, Cursus mathematicus (Wiirzburg, 1661), p. 576.
2 Ad Maiorem Dei Gloriam Geometriae speciosae Elementa, .... Pctri Mengoli
(Bologna, 1659), p. 20.
3 Michaelis Angeli Riccii Exercitatio geometrica (Londini, 1668), p. 2. [Preface,
1666.]
4 Journal des 8f avans, l'anne*e 1680 (Amsterdam, 1682), p. 160.
6 Ibid., Tanned 1684, Vol. II (2d ed.; Amsterdam, 1709), p. 254.
6 Philosophical Transactions, Vol. XV-XVI (London, 1684-91), p. 189.
7 R. P. Claudii Francisci Milliet Dechales Camberiensis Mundus rnathematicus,
Tomus tertius (Leyden, 1674), p. 664; Tomus primus (editio altera; Leyden,
1690), p. 635.
8 Andreas Spole, Arilhmelica vutgaris et specioza (Upsala, 1692). See G. Ene-
strom in L'Inlermediaire des mathtmaticiens, Vol. IV (1897), p. 60.
9 Joseph Moxon, Mathematical Dictionary (London, 1701), p. 190, 191.
10 (Euvres de Descartes, Vol. V (1903), p. 506, 507.
POWERS 349
indicate how the science might have been retarded in its progress
under the handicap of cumbrous notations, had such wise leadership
as that of Descartes, Wallis, and Newton not been available. Rich.
Balarn1 in 1653 explains a device of his own, as follows: "(2) j 3 \ , the
Duplicat, or Square of 3, that is, 3X3; (4) • 2 • , the Quaclruplicat of 2,
that is, 2X2X2X2-16." The Dutch J. Stampioen2 in 1639 wrote
DA for A2; as early as 1575 F. Maurolycus3 used D to designate the
square of a line. Similarly, an Austrian, Johannes Cararnuel,4 in
1670 gives "Q25. est Quadratum Numeri 25. hoc est, 625."
Huygens5 wrote "1000(3)10" for 1,000 = 103, and "1024(10)2" for
1024 ==210. A Leibnizian symbolism6 explained in 1710 indicates the
cube of AB+BC thus: [3] (AB+BC); in fact, before this time, in
1695 Leibniz7 wrote \m\ y+a for (y+a)m.
304. Descartes preferred the notation aa to a2. Fr. van Schooten,8
in 1646, followed Descartes even in writing <jqy xx rather than </2, x2,
but in his 1649 Latin edition of Descartes' geometry he wrote prefer-
ably x2. The symbolism xx was used not only by Descartes, but also
by Huygens, Hahn, Kersey, Wallis, Newton, Halley, Rolle, Euler —
in fact, by most writers of the second half of the seventeenth and of
the eighteenth centuries. Later, Gauss9 was in the habit of writing
xx, and he defended his practice by the statement that x2 did not take
up less space than xx, hence did not fulfil the main object of a symbol.
The x* was preferred by Leibniz, Ozanam, David Gregory, and Pascal.
305. The reader should be reminded at this time that the repre-
sentation of positive integral powers by the repetition of the factors
was suggested very early (about 1480) in the Dresden Codex C. 80
under the heading Algorithmus de additis et minutis where x2 — z and
xw — zzzzz; it was elaborated more systematically in 1553 by M.
1 Rich. Balam, Algebra, or The Doctrine of Composing, Inferring, and tie-
solving an Equation (London, 1653), p. 9.
2 Johan Stampioen, Algebra (The Hague, 1639), p. 38.
3 D. Francisc' Mavrolyci Abbatis measancnsis Opuscula Malhcmatica (Venice,
1575) (Euclid, Book XIII),- p. 107.
4 Joannis Caramvelis Malhesis Biceps. Veins et Nova (Companiae, 1070),
p. 131, 132.
5 Christ iani Hugenii Opera .... quae collegit .... Guilielmus Jacobus's
Gravesande (Ley den, 1751), p. 456.
0 Miscellanea Berolinensia (Berlin, 1710), p. 157.
7 Acta eruditorum (1695), p. 312.
8 Francisci a Schooten Leydensis de Organica conicarum sectionum .... Trac-
talus (Leyden, 1646), p. 91 ff.
9 M. Cantor, op. cit., Vol. II (2d ed.), p. 794 n.
350 A HISTORY OF MATHEMATICAL NOTATIONS
Stifel (§ 156). One sees in Stifcl the exponential notation applied,
not to the unknown but to several different quantities, all of them
known. Stifel understood that a quantity with the exponent zero
had the value 1. But this notation was merely a suggestion which
Stifel himself did not use further. Later, in Alsted's Encyclopaedia,1
published at Herborn in Prussia, there is given an explanation of the
German symbols for radix, zensus, cubus, etc. ; then the symbols from
Stifel, just referred to, are reproduced, with the remark that they are
preferred by some writers. The algebra proper in the Encyclopaedia
is from the pen of Johann Geysius2 who describes a similar notation
2a, 4aa, 8aaa, . . . . , 512aaaaaaaaa and suggests also the use of
I II III IX
Roman numerals as indices, as in 21 4q Sc . . . . 512cc. Forty years
after, Caramvel3 ascribes to Geysius the notation aaa for the cube of
a, etc.
In England the repetition of factors for the designation of powers
was employed regularly in Thomas Harriot. In a manuscript pre-
served in the library of Sion College, Nathaniel Torporley (1573-
1632) makes strictures on Harriot's book, but he uses Harriot's
notation.4 John Newton5 in 1654 writes aaaaa. John Collins writes
in the Philosophical Transactions of 1668 aaa— -3aa+4a = jV to signify
a;3— 3z2+4:c = jV. Harriot's mode of representation is found again in
the Transactions6 for 1684. Joseph Raphson7 uses powers of g up to
010, but in every instance he writes out each of the factors, after the
manner of Harriot.
306. The following curious symbolism was designed in 1659 by
Samuel Foster8 of London :
~1 -I =1 ID =1 3 3 ID
q c qq qc cc qqc qcc ccc
2 345 6789
1 Johannis-Henrici Alstedii Encyclopaedia (Herborn, 1630), Book XIV,
"Arithmetica, " p. 844.
2 Ibid., p. 865-74.
3 Joannis Caramvelis Mathesis Biceps (Campaniae, 1670), p. 121.
4 J. O. Halliwcll, A Collection of Letters Illustrative of the Progress of Science in
England (London, 1841), p. 109-16.
5 John Newton, Institutio Mathematica or a Mathematical Institution (London,
1654), p. 85.
6 Philosophical Transactions, Vol. XV-XVI (London, 1684-91), p. 247, 340.
7 Josepho Raphson, Analysis Aequationum universalis (London, 1702).
[First edition, 1697.]
8 Samuel Foster, Miscellanies, or Mathematical Lucubrations (London, 1659),
!>. 10.
POWERS 351
Foster did not make much use of it in his book. He writes the pro-
portion
"At AC . AR::CD\:'RP\999
which means
An altogether different and unique procedure is encountered in
the Maandelykse Mathematische Liefhebberye (1754-69), where \/ -
signifies extracting the mth root, and - \/ signifies raising to the
rath power. Thus,
307. Spread of Descartes' notation. — Since Descartes' Geometric
appeared in Holland, it is not strange that the exponential notation
met with prompter acceptance in Holland than elsewhere. We have
already seen that J. Stampioen used this notation in 1639 and 1640.
The great disciple of Descartes, Fr. van Schooten, used it in 1646, and
in 1649 in his Latin edition of Descartes' geometry. In 1646 van
Schooten indulges1 in the unusual practice of raising some (but not all)
of his coefficients to the height of exponents. He writes #3 — *aax — 2a3 »
0 to designate a;3 — 3a2£— 2a3 = 0. Van Schooten2 does the same thing
in 1657, when he writes 2ax for 2ax. Before this Marini Ghetaldi3
in Italy wrote coefficients in a low position, as subscripts, as in the
proportion,
" ut AQ ad A2 in B ita - ad m<>-\-n ,"
y
Cf
which stands for A*:2AB= :(2m+ri). Before this Albert Girard4
placed the coefficients where we now write our exponents. I quote:
"Soit un binome conjoint B+C. Son Cube era B(Bq+C$) +
)." Here the cube of B+C is given in the form corresponding
1 Francisci a Schooten, De organica conicarum sedionum .... tractatus
(Leyden, 1646), p. 105.
2 Francisci a Schooten, Exerdtationum mathematicarum liber primus (Leyden,
1657), p. 227, 274, 428, 467, 481, 483.
3 Marini Ghetaldi, De resolutions et compositione mathematica libri quinque.
Opus posthumum (Rome, 1630). Taken from E. Gelcich, Abhandlungen zur Ge»
schichte der Mathematik, Vol. IV (1882), p. 198.
4 A. Girard, Invention nouvelle en I'algebre (1629), "3 C."
352 A HISTORY OF MATHEMATICAL NOTATIONS
to£(£2+3C2)+C(3£2+C2). Much later, in 1679, we find in the col-
lected works of P. Fermat1 the coefficients in an elevated position:
2D in A for 2DA, 2R in E for 2RE.
The Cartesian notation was used by C. Huygens and P. Mersenne
in 1646 in their correspondence with each other,2 by J. Hudde3 in
1658, and by other writers.
In England, J. Wallis4 was one of the earliest writers to use
Descartes' exponential symbolism. He used it in 1655, even though he
himself had been trained in Oughtred's notation.
The Cartesian notation is found in the algebraic parts of Isaac
Barrow's5 geometric lectures of 1670 and in John Kersey's Algebra^ of
1673. The adoption of Descartes7 a4 in strictly algebraic operations
and the retention of the older Aq, Ac for A2, A3 in geometric analysis is
of frequent occurrence in Barrow and in other writers. Seemingly,
the impression prevailed that A2 and A3 suggest to the pupil the purely
arithmetical process of multiplication, A A and A A A, but that the
symbolisms Aq and Ac conveyed the idea of a geometric square and
geometric cube. So we find in geometrical expositions the use of the
latter notation long after it had disappeared from purely algebraic
processes. We find it, for instance, in W. Whiston's edition of
Tacquet's Euclid)7 in Sir Isaac Newton's Principia8 and Opticks* in B.
Robins7 Tracts,10 and in a text by K. F. Hauber.11 In the Philosophical
Transactions of London none of the pre-Cartesian notations for powers
appear, except a few times in an article of 1714 from the pen of K.
Cotes, and an occasional tendency to adhere to the primitive, but very
1 P. Fermat, Varia opera (Toulouse, 1679), p. 5.
2 C. Iluygens, (Euvres, Vol. I (La Haye, 1888), p. 24.
3 Joh. Huddeni Epist. I de reductione aequatwnum (Amsterdam, 1658);
Matthicsscn, Grundzuge der Antiken u. Modcrnen Algebra (Leipzig, 1878), p. 349.
4 John Wallis, Arilhmetica infinitorurn (Oxford, 1655), p. 16 ff.
5 Isaac Barrow, Lectiones Geometriae (London, 1670), Lecture XIII (W.
Whcwell's ed.), p. 309.
6 John Kersey, Algebra (London, 1673), p. 11.
7 See, for instance, Elernenta Euclidea geomelriae auctore Andrea Tacquet, ...
Gulielmus Whiston (Amsterdam, 1725), p. 41.
8 Sir Isaac Newton, Principia (1687), Book I, Lemma xi, Cas. 1, and in other
places.
9 Sir Isaac Newton, Opticks (3d ed.; London, 1721), p. 30.
10 Benjamin Robins, Mathematical Tracts (ed. James Wilson, 1761), Vol. II,
p. 65.
11 Karl Friderich Hauber, Archimeds zwey Biicher ilber Kugel und Cylinder
(Tubingen, 1798), p. 56 ff.
POWERS 353
lucid method of repeating the factors, as aaa for a3. The modern
exponents did not appear in any of the numerous editions of William
Oughtrcd's Clavis mathematicae; the last edition of that popular book
was issued in 1694 and received a new impression in 1702. On Febru-
ary 5, 1666-67, J. Wallis1 wrote to J. Collins, when a proposed new
edition of Oughtred's Clavis was under discussion: "It is true, that as
in other things so in mathematics, fashions will daily alter, and that
which Mr. Oughtred designed by great letters may be now by others
designed by small; but a mathematician will, with the same ease and
advantage, understand Ac or aaa." As late as 1790 the Portuguese
J. A. da Cunha2 occasionally wrote Aq and Ac. J. Pell wrote r2 and t2
in a letter written in Amsterdam on August 7, 1645.3 J. H. Rahn's
Teutsche Algebra, printed in 1659 in Zurich, contains for positive inte-
gral powers two notations, one using the Cartesian exponents, a8, x4,
the other consisting of writing an Archemidean spiral (Fig. 96) be-
tween the base and the exponent on the right. Thus a© 3 signifies
a8. This symbol is used to signify involution, a process which Rahn
calls involviren. In the English translation, made by T. Brancker and
published in 1668 in London, the Archimedean spiral is displaced by
the omicron-sigrna (Fig. 97), a symbol found among several English
writers of textbooks, as, for instance, J. Ward,4 E. Hatton,5 Ham-
mond,6 C. Mason,7 and by P. Ronayne* — all of whom use also llahn's
and Brancker 's ww' to signify evolution. The omicrori-sigma is
found in Birks;9 it is mentioned by Saverien,10 who objects to it as
being superfluous.
Of interest is the following passage in Newton's Arithmetick,11
which consists of lectures delivered by him at Cambridge in the period
1669-85 and first printed in 1707: "Thus ^64 denotes 8; and i/3:64
1 Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. I
(Oxford, 1841), p. 63.
2 J. A. da Cunha, Principios mathematicos (1790), p. 158.
3 J. O. Halliwell, Progress of Science in England (London, 1841), p. 89.
4 John Ward, The Young Mathematician's Guide (London, 1707), p. 144.
6 Edward Hatton, Inlire System of Arithmetic (London, 1721), p. 287.
6 Nathaniel Hammond, Elements of Algebra (London, 1742).
7 C. Mason in the Diarian Repository (London, 1774), p. 187.
8 Philip Ronayne, Treatise of Algebra (London, 1727), p. 3.
9 Anthony and John Birks, Arithmetical Collections (London, 1766), p. viii.
10 A. Saverien, Dictionnaire universel de mathematique et de physique (Paris,
1753), "Caractere."
11 Newton's Universal Arithmetick (London, 1728), p. 7.
354 A HISTORY OF MATHEMATICAL NOTATIONS
denotes 4 There are some that to denote the Square of the first
Power, make use of <?, and of c for the Cube, qq for the Biquadrate,
and qc for the Quadrato-Cube, etc Others make use of other
sorts of Notes, but they are now almost out of Fashion."
In the eighteenth century in England, when parentheses were
seldom used and the vinculum was at the zenith of its popularity,
bars were drawn horizontally and allowed to bend into a vertical
" IT
stroke1 (or else were connected with a vertical stroke), as in AXBU\
or in a"+"b|n"'
In France the Cartesian exponential notation was not adopted as
early as one might have expected. In J. de Billy's Nova geometriae
clavis (Paris, 1643), there is no trace of that notation; the equation
z+z2 = 20 is written "Ifl+lQ aequatur 20." In Fermat's edition2 of
Diophantus of 1670 one finds in the introduction 1QQ+4C+10Q+
2(W+1 for xt+4x*+10x*+20x+l. But in an edition of the works of
Fermat, brought out in 1679, after his death, the algebraic notation
of Vieta which he had followed was discarded in favor of the expo-
nents of Descartes.3 B. Pascal4 made free use of positive integral
exponents in several of his papers, particularly the Potestatum numeri-
carum summa (1654).
In Italy, C. Renaldini5 in 1665 uses both old and new exponential
notations, with the latter predominating.
308. Negative, fractional, and literal exponents. — Negative and
fractional exponential notations had been suggested by Oresme,
Chuquet, Stevin, and others, but the modern symbolism for these is
due to Wallis and Newton. Wallis6 in 1656 used positive integral
exponents and speaks of negative and fractional "indices," but he
does not actually write a~l for -, a3 for >/a3. He speaks of the series
1 See, for instance, A. Malcolm, A New System of Arithmetick (London, 1730),
p. 143.
2 Diophanti Alexandrini arithmeticarum Libri Sex, cum commentariis G. B.
Bacheti V. C. et observationibus D. P. de Fermat (Tolosae, 1670), p. 27.
3 See (Euvres de Fermat (e*d. Paul Tannery et Charles Henry), Tome I (Paris,'
1891), p. 91 n.
4 (Euvres de Pascal (e"d. Leon Brunschvicg et Pierre Boutroux), Vol. Ill (Paris,
1908), p. 349-58.
5 Caroli Renaldinii, Ars analytica .... (Florence, 1665), p. 11, 80, 144.
6 J. Wallis, Arithmetica infinitorum (1656), p. 80, Prop. CVI.
POWERS 355
_7=> „ _^^ e^c ^ as having the "index — |." Our modern notation
F 1 1/2 "K 3
involving fractional and negative exponents was formally introduced
a dozen years later by Newton1 in a letter of June 13, 1676, to Olden-
burg, then secretary of the Royal Society of London, which explains
the use of negative and fractional exponents in the statement, "Since
algebraists write a2, a3, a4, etc., for aa, aaa, aaaa, etc., so I write a*,
a3, a4, for I/a, I/a3, 1/c a5; and I write a~l, a~2, a~3, etc., for -, - , — ,
etc." He exhibits the general exponents in his binomial formula first
announced in that letter:
F+PQl- -PM AQ+m~- n- BQ+™-2? CQ+™-*? DQ+ , etc.,
H £11 o/l 4?1»
m m m m
where A=Pn, B = - PnQ} etc., and where — may represent any real
it ill
and rational number. It should be observed that Newton wrote here
literal exponents such as had been used a few times by Wallis,2 in
1657, in expressions like i/dRd = R, ARmXARn = A2Rm+n, which arose
in the treatment of geometric progression. Wallis gives also the
division ARm) ARm+n(Rn. Newton3 employs irrational exponents in his
letter to Oldenburg of the date October 24, 1676, where he writes
x^2+xv/7\ a =y. Before Wallis and Newton, Victa indicated general
exponents a few times in a manner almost rhetorical ;4 his
A , , , E potestate — A potesta . . ,
A potestas-] — ^ , -.—- A- — , — in A gradum
^ E gradui+A gradu J
is our
the two distinct general powers being indicated by the words potestas
and gradus. Johann Bernoulli5 in 1691-92 still wrote 3D frax+xx for
1 Isaaci Newtoni Opera (cd. S. Horsely), Tom. IV (London, 1782), p. 215.
2 J. Wallis, Mathesis universalis (Oxford, 1657), p. 292, 293, 294.
3 See J. Collins, Commercium epistolicum (ed. J. B. Biot and F. Lefort; Paris,
1856), p. 145.
4 Vieta, Opera mathematica (ed. Fr. van Schootcn, 1634), p. 197.
5 lohannis I Bernoulli, Lecliones de calculo differentialium .... von Paul
Schafheitlin. Scparatabdruck aus den V erhandlungen der Naturforschenden Ge-
sellschaft in Basel, Vol. XXXIV, 1922.
356 A HISTORY OF MATHEMATICAL NOTATIONS
iCi/yx+xx for 4v//(t/x+x2)3, 5QQl/ayx+x3+zyx for
But fractional, negative, and general exponents
were freely used by D. Gregory1 and were fully explained by W. Jones2
and by C. Reyneau.3 Reyneau remarks that this theory is not ex-
plained in works on algebra.
309. Imaginary exponents. — The further step of introducing im-
aginary exponents is taken by L. Euler in a letter to Johann Ber-
noulli,4 of October 18, 1740, in which he announces the discovery of
the formula c+x|/~1+e~x»/~1 = 2 cos x, and in a letter to C. Gold-
bach,6 of December 9, 1741, in which he points out as a curiosity
-
that the fraction - - - is nearly equal to } J. The first ap-
z
pearance of imaginary exponents in print is in an article by Euler
in the Miscellanea Berolinensia of 1743 and in Eulcr's Introduetio in
analysin (Lausannae, 1747), Volume I, page 104, where he gives the
all-important formula e+1)|//~1 = cos v+l/— 1 sin v.
310. At an earlier date occurred the introduction of variable
exponents. In a letter of 1679, addressed to C. Huygens, G. W.
Leibniz6 discussed equations of the form xx— z=24,£s-f-2* = byxx+zz = c.
On May 9, 1694, Johann Bernoulli7 mentions expressions of this
sort in a letter to Leibniz who, in 1695, again considered exponentials
in the Acta eruditorum, as did also Johann Bernoulli in 1697.
311. Of interest is the following quotation from a discussion by
T. P. Nunn, in the Mathematical Gazette, Volume VI (1912), page
255, from which, however, it must not be inferred that Wallis
actually wrote down fractional and negative exponents: "Those
who are acquainted with the work of John Wallis will remember
that he invented negative and fractional indices in the course of
an investigation into methods of evaluating areas, etc. He had
1 David Gregory, Exercitatio geometrica de dimensione ftgurarum (Edinburgh,
1684), p. 4-6.
2 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 67,
115-19.
3 Charles Reyneau, Analyse demontree (Paris, 1708), Vol. I, Introduction.
4 See G. Enestrom, BiUiotheca mathematica (2d ser.), Vol. XI (1897), p. 49.
8 P. H. Fuss, Correspondance mathematique et physique (Petersburg, 1843),
Vol. I, p. 111.
6C. I. Gerhardt, Brief wechsel von G. W. Leibniz mil Mathematikern (2d ed.;
Berlin, 1899), Vol. I, p. 568.
7 Johann Bernoulli in Leibnizens Malhematische Schriften (ed. C. I. Gerhardt),
Vol. Ill (1855), p. 140.
POWERS 357
discovered that if the ordinates of a curve follow the law y~kxn its
area follows the law -A=— T-T ' kxn+l, n being (necessarily) a positive
integer. This law is so remarkably simple and so powerful as a method
that Wallis was prompted to inquire whether cases in which the ordi-
k
nates follow such laws as y — — y = k \/x could not be brought within
x ,
its scope. He found that this extension of the law would be possible
k ~
if — could be written kx~", and k \/x as kxn. From this, from numerous
*c
other historical instances, and from general psychological observa-
tions, I draw the conclusion that extensions of notation should be
taught because arid when they are needed for the attainment of some
practical purpose, and that logical criticism should come after the
suggestion of an extension to assure us of its validity."
312. Notation for principal values. — When in the early part of the
nineteenth century the multiplicity of values of an came to be studied,
where a and n may be negative or complex numbers, and when the
need of defining the principal values became more insistent, new nota-
tions sprang into use in the exponential as well as the logarithmic
theories. A. L. Cauchy1 designated all the values that an may take,
for given values of a and n [a^O], by the symbol ((a))*, so that ((a))*^
exiat€2kx*if wncre i means the tabular logarithm of |a|, e = 2.718 . . . . ,
7r = 3.141 . . . . , fc = 0, ±1, ±2, .... This notation is adopted by
O. Stolz and J. A. Gmeiner2 in their Theoretische Arithmetik.
Other notations sprang up in the early part of the last century,
Martin Ohm elaborated a general exponential theory as early as 1821
in a Latin thesis and later in his System der Mathematik ( 1822-33). 3
In ax, when a and x may both be complex, log a has an infinite num-
ber of values. When, out of this infinite number some particular
value of log a, say <x , is selected, he indicates this by writing (a|| « ).
With this understanding he can write x log a-\-y log a= (x-\-y) log a,
and consequently ax<ay = ax+y is a complete equation, that is, an equa-
tion in which both sides have the same number of values, representing
exactly the same expressions. Ohm did not introduce the particular
value of ax which is now called the "principal value."
1 A. L. Cauchy, Cours d' analyse (Paris, 1821), chap, vii, § 1.
2 O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902),
p. 371-77.
3 Martin Ohm, Vcrsuch eines vollkommen consequcnten Systems der Mathe-
matik, Vol. II (2d ed., 1829), p. 427. [First edition of Vol. II, 1823.]
358 A HISTORY OF MATHEMATICAL NOTATIONS
Crelle1 let \u\k indicate some fixed value of uk, preferably a real
value, if one exists, where k may be irrational or imaginary; the two
vertical bars were used later by Weierstrass for the designation of
absolute value (§ 492).
313. Complicated exponents. — When exponents themselves have
exponents, and the latter exponents also have exponents of their own,
then clumsy expressions occur, such as one finds in Johann I Ber-
noulli,2 Goldbach,3 Nikolaus II Bernoulli,4 and Waring.
2. Sit data exponentialis quantitas *' x v, & per przecedentem
methodum inveniri poteft cjus fluxio **v 4* vytf~*x •+. v x & x log.
xxy.
3. Sit exponentialis quantitas & , & ejus fluxio erit y* x x*
Ac, Arc,
4P *€• W ft€»
v &c. w v w fte.
* v t» &C*
x z -f- ^ xj>* x fc* x w*c x vv •*' x log. ^ x log,^ x log.
te
» *» *e,
» v tp &e.
ZXV-f-*" X^" X5J* XV^X &C. W**"-"1 X log. ^ X log. y X.
log. 2 x log. v x w 4- &c* unde facile conftabit lex, quam obfervat
fcries* _
FIG. 106. — E. Waring's ''repeated exponents." (From Meditationes analyticae
[1785], p. 8.)
De Morgan5 suggested a new notation for cases where exponents
are complicated expressions. Using a solidus, he proposes a A { (a-\-bx)
/(c+ex)}j where the quantity within the braces is the exponent of a.
He returned to this subject again in 1868 with the statement: "K
convenient notation for repeated exponents is much wanted: not a
working symbol, but a contrivance for preventing the symbol from
wasting a line of text. The following would do perfectly well, #|a|&|c|d,
1 A. L. Crelle in Crelle's Journal, Vol. VII (1831), p. 265, 266.
2 lohann I Bernoulli, Ada eruditorum (1697), p. 125-33.
» P. II. Fuss, Correspondence math, et phys. ... du XVIII* siecle, Vol. II (1843),
p. 128.
4 Op. cit., p. 133.
6 A. de Morgan, "Calculus of Functions," Encyclopaedia Metropolitana, Vol.
II (1845), p. 388.
POWERS 359
in which each post means all which follows is to be placed on the top
of it. Thus:1
x\a\b\c\d = x^c\d = x*blcld = ^6c|d = x^ ."
When the base and the successive exponents are all alike, say a,
Woepcke2 used the symbol Q for((aa)a"\a and ^ for a(a"(°^) where
m indicates the number of repetitions of a. He extended this notation
to cases where a is real or imaginary, not zero, and m is a positive
or negative integer, or zero. A few years later J. W. L. Glaisher sug-
gested still another notation for complicated exponents, namely,
1 T
atxM — -f, the arrows merely indicating that the quantity between
x
them is to be raised so as to become the exponent of a. Glaisher prefers
this to "a Exp. u" for au. Harkness and Morley3 state, "It is usual
to write exp (z)=ez, when z is complex." The contraction "exp" was
recommended by a British Committee (§ 725) in 1875, but was ignored
in the suggestions of 1916, issued by the Council of the London Math-
ematical Society. G. H. Bryan stresses the usefulness of this symbol.4
Another notation was suggested by H. Schubert. If a° is taken as
an exponent of a, one obtains a^ or aa°, and so on. Schubert desig-
nates the result by (a; 6), indicating that a has been thus written 6
times.5 For the expression (a;6)(tt;c) there has been adopted the sign
(a; 6+c), so that (a; &)<<" c> = (a; c+l)<°: *-».
314. D. F. Gregory6 in 1837 made use of the sign (+)f, r an integer,
to designate the repetition of the operation of multiplication. Also,
(+a2)i = +i(a2)i=+*a> where the +* "will be different, according as
we suppose the + to be equivalent to the operation repeated an even or
an odd number of times. In the former case it will be equal to +, in
the latter to — . And generally, if we raise +a to any power m,
whether whole or fractional, we have (+a)m = +mam So long as
1 A. de Morgan, Transactions of the Cambridge Philosophical Society, Vol. XI,
Part III (1869), p. 450.
2 F. Woepcke in Crelle's Journal, Vol. XLII (1851), p. 83.
3 J. Harkness and F. Morley, Theory of Functions (New York, 1893), p. 120.
4 Mathematical Gazette, Vol. VIII (London, 1917), p. 172, 220.
6 H. Schubert in Encyclopedic d. scien. math., Tome I, Vol. I (1904), p. 61.
L. Euler considered a°Py etc. See E. M. Le"meray, Proc. Edinb. Math. Soc., Vol.
XVI (1897), p. 13.
8 The Mathematical Writings of Duncan Farquharson Gregory (ed. William
Walton; Cambridge, 1865), p. 124-27, 145.
360 A HISTORY OF MATHEMATICAL NOTATIONS
m is an integer, nn is an integer, and +rmam has only one value; but if
v r-
m be a fraction of the form -, + « will acquire different values, ac-
cording as we assign different values to r .... i/( — a)Xj/(~a) =
l/(+a2) — ]/(-f )|/(a2) = —a; for in this case we know how the +
has been derived, namely from the product — — = +, or — 2=+,
which of course gives + * = — , there being here nothing indeterminate
about the +. It was in consequence of sometimes tacitly assuming
the existence of +, and at another time neglecting it, that the errors
in various trigonometrical expressions arose; and it was by the intro-
duction of the factor cos 2r?H — * sin 2rv (which is equivalent to +r)
that Poinsot established the formulae in a more correct and general
P p
shape." Gregory finds "sin (+qc) = +<* sin c."
A special notation for the positive integral powers of an imaginary
root r of xn~l+xn~2+ .... +x+l = 0, n being an odd prime, is given
by Gauss;1 to simplify the typesetting he designates r, rr, r3, etc.,
by the symbols [1], [2], [3], etc.
315. Conclusions. — There is perhaps no symbolism in ordinary
algebra which has been as well chosen and is as elastic as the Cartesian
exponents. Descartes wrote a3, z4; the extension of this to general
exponents a" was easy. Moreover, the introduction of fractional and
negative numbers, as exponents, was readily accomplished. The ir-
rational exponent, as in a1/2, found unchallenged admission. It was
natural to try exponents in the form of pure imaginary or of complex
numbers (L. Euler, 1740). In the nineteenth century valuable inter-
pretations were found which constitute the general theory of bn
where b and n may both be complex. Our exponential notation has
been an aid for the advancement of the science of algebra to a degree
that could not have been possible under the old German or other early
notations. Nowhere is the importance of a good notation for the rapid
advancement of a mathematical science exhibited more forcibly than
in the exponential symbolism of algebra.
SIGNS FOR ROOTS
316. Early forms. — Symbols for roots appear very early in the
development of mathematics. The sign [p'for square root occurs in two
Egyptian papyri, both found at Kahun. One was described by F. L.
1 C. F. Gauss, Disquisitions arithmeticae (Leipzig, 1801), Art. 342; Werke,
Vol. I (1863), p. 420.
ROOTS 361
Griffith1 and the other by H. Schack-Schackenburg.2 For Hindu signs
see §§ 107, 108, 112; for Arabic signs see § 124.
317. General statement. — The principal symbolisms for the desig-
nation of roots, which have been developed since the influx of Arabic
learning into Europe in the twelfth century, fall under four groups
having for their basic symbols, respectively, R (radix), I (latus), the
sign |/, and the fractional exponent.
318. The sign I}; first appearance. — In a translation8 from the
Arabic into Latin of a commentary of the tenth book of the Elements
of Euclid, the word radix is used for "square root." The sign R came
to be used very extensively for "root/' but occasionally it stood also
for 'the first power of the unknown quantity, x. The word radix was
used for x in translations from Arabic into Latin by John of Seville
and Gerard of Cremona (§ 290). This double use of the sign R for x
and also for square root is encountered in Leonardo of Pisa (§§122,
292)4 and Luca Pacioli (§§ 135-37, 292).
Before Pacioli, the use of R to designate square root is also met in a
correspondence that the German astronomer Regiomontanus (§ 126)5
carried on with Giovanni Bianchini, who was court astronomer at
Ferrara in Italy, and with Jacob von Speier, a court astronomer at
Urbino (§ 126).
In German manuscripts referred to as the Dresden MSS C. 80,
written about the year 1480, and known to have been in the hands of
J. Widman, H. Grammateus, and Adam Riese, there is a sign con-
sisting of a small letter, with a florescent stroke attached (Fig. 104).
It has been interpreted by some writers as a letter r with an additional
stroke. Certain it is that in Johann Widman's arithmetic of 1489
occurs the crossed capital letter R, and also the abbreviation ra
(§293).
Before Widman, the Frenchman Chuquet had used R for "root"
1 F. L. Griffith, The Petrie Papyri, I. Kahun Papyri, Plate VIII.
2 H. Schack-Schackenburg in Zeitschrift fur aegyptische Sprache und Alter-
tumskunde, Vol. XXXVIII (1900), p. 136; also Plate IV. See also Vol. XL,
p. 65.
3 M. Curtzo, Anaritii in decent libros elementorum Euclidis commentarii (Leipzig,
1899), p. 252-386.
4 Scritti di Leonardo Pisano (ed. B. Boncompagni), Vol. II (Rome, 1862),
"La practica geometriae," p. 209, 231. The word radix, meaning x, is found also
in Vol. I, p. 407.
6 M. Curtze, "Der Briefwechsel Riomantan's, etc.," Abhandlungen zur Ge-
schichte der mathematischen Wissenschaften, Vol. XII (Leipzig, 1902), p. 234,
318.
362 A HISTORY OF MATHEMATICAL NOTATIONS
in his manuscript, Le Triparty (§ 130). He1 indicates ft* 16. as 4,
"ft*. 16. si est . 2.," "#5. 32. si est . 2."
319. Sixteenth-century use of ft. — The different uses of ft. made in
Pacioli's Summa (1494, 1523) are fully set forth in §§ 134-38. In
France, De la Roche followed Chuquet in the use of ft. (§ 132). The
symbol appears again in Italy in Ghaligai's algebra (1521), and in later
editions (§ 139), while in Holland it appeared as early as 1537 in the
arithmetic of Giel Van der Hoecke (§ 150) in expressions like "Item
wildi aftrecken ft | van ft | resi ft -J"; i.e., V\ — V~\ = V\. The em-
ployment of ft in the calculus of radicals by Cardan is set forth in
§§ 141, 199. A promiscuous adoption of different notations is found in
the algebra of Johannes Scheubel (§§ 158, 159) of the University of
Tubingen. He used Widman's abbreviation ra, also the sign j/i he
indicates cube root by ra. cu. or by /vw/> fourth root by ra. ra. or by
/W/. He suggests a notation of his own, of which he makes no further
use, namely, radix se., for cube root, which is the abbreviation of radix
secundae quantitatis. As the sum "ra. 15 ad ra. 17" he gives "ra. col.
32+1/1020," i.e., 1/15 +1/17 = v"32+ 1/1020. The col, collecti,
signifies here aggregation.
Nicolo Tartaglia in 1556 used ft extensively and also parentheses
(§§ 142, 143). Francis Maurolycus2 of Messina in 1575 wrote "r. 18"
for 1/18, "r. v.Qm.r 7£" for 1/6 — 1/7J. Bombelli's radical notation
is explained in § 144. It thus appears that in Italy the ft had no rival
during the sixteenth century in the calculus of radicals. The only
variation in the symbolism arose in the marking of the order of the
radical and in the modes of designation of the aggregation of terms
that were affected by ft.
320. In Spain3 the work of Marco Aurel (1552) (§ 204) employs
the signs of Stifel, but Antich Rocha, adopting the Italian abbrevia-
tions in adjustment to the Spanish language, lets, in his Arithmetica
of 1564, "15 Mas ra. q. 50 Mas ra. q. 27 Mas ra. q. 6" stand for
15+1/50+ 1/27+1/6. A few years earlier, J. Perez de Moya, in his
Aritmetica practica y speculativa (1562), indicates square root by r,
1 Le Triparty en la science des nombres par Maistre Nicolas Chuquet Parisien ...
par M. Aristide Marre in Boncompagni's Bullettino, Vol. XIII, p. 655; (reprint,
Rome, 1881), p. 103.
2 D. Francisci Mavrolyci, Abbatis Messanensi8t Opuscula mathematica (Venice,
1575), p. 144.
3 Our information on these Spanish authors is drawn partly from Julio Rey
Pastor, Los Matemdticos espanoles de siglo XVI (Oviedo, 1913), p. 42.
ROOTS 363
cube root by rrr, fourth root by rr, marks powers by eo., ce., cu., c. ce.,
and "plus" by p, "minus" by ra, "equal" by eq.
In Holland, Adrianus Romanus1 used a small r, but instead of v
wrote a dot to mark a root of a binomial or polynomial; he wrote
r bin. 2+r bin. 2+r bin. 2+r 2. to designate
In Tartaglia's arithmetic, as translated into French by Gosselin2
of Caen, in 1613, one finds the familiar /? cu to mark cube root. A
modification was introduced by the Scotsman James Hume,3 residing
in Paris, who in his algebra of 1635 introduced Roman numerals to
indicate the order of the root (§ 190). Two years later, the French
text by Jacqves de Billy4 used 5Q, &C, RQC for v7"", V ~~, i/~~>
respectively.
321. Seventeenth-century use of R. — During the seventeenth cen-
tury, the symbol R lost ground steadily but at the close of the century
it still survived; it was used, for instance, by Michael Rolle5 who em-
ployed the signs 2+R. — 121. to represent 2-fV7 — 121, and R. trin.
6oa66-9a46-63 to represent i/6a262 -- 9a46 -- 68~. In 1690 H. Vitalis6
takes Ri to represent secunda radix, i.e., the radix next after the square
root. Consequently, with him, as with Scheubel, 3. R. 2* 8, meant
3fx/8, or 6.
The sign R or #, representing a radical, had its strongest foothold
in Italy and Spain, and its weakest in England. With the close of the
seventeenth century it practically passed away as a radical sign; the
symbol \/ gained general ascendancy. Elsewhere it will be pointed
out in detail that some authors employed R to represent the unknown
x. Perhaps its latest regular appearance as a radical sign is in the
Spanish text of Perez de Moya (§ 320), the first edition of which ap-
peared in 1562. The fourteenth edition was issued in 1784; it still
gave rrr as signifying cube root, and rr as fourth root. Moya's book
offers a most striking example of the persistence for centuries of old and
clumsy notations, even when far superior notations are in general use.
1 Ideae Mathematical Pars Prima, .... Adriano Romano Lovaniensi (Ant-
werp, 1593), following the Preface.
2 U Arithmetique de Nicolas Tartaglia Brescian, traduit ... par Gvillavmo
Gosselin de Caen, Premier Partie (Paris, 1613), p. 101.
3 James Hume, Traite de Valgebre (Paris, 1635), p. 53.
4 Jacqves de Billy, Abrege des Preceples d'Algebre (Rheims, 1637), p. 21.
6 Journal des Scavans de TAn 1683 (Amsterdam, 1709), p. 97.
6 Lexicon mathematicum ... authore Hieronymo Vitali (Rome, 1690), art.
"Algebra."
364 A HISTORY OF MATHEMATICAL NOTATIONS
322. The sign 1. — The Latin word latus ("side of a square") was
introduced into mathematics to signify root by the Roman surveyor
Junius Nipsus,1 of the second century A.D., and was used in that sense
by Martianus Capella,2 Gerbert,3 and by Plato of Tivoli in 1145, in
his translation from the Arabic of the Liber embadorum (§ 290).
The symbol I (latus) to signify root was employed by Peter Ramus4
with whom "I 27 ad I 12" gives "I 75," i.e., 1/27+1/12 = 1/76;
"II 32 de II 162" gives "tt 2," i.e., 1/32 from I7 162= ^2. Again,5
"8-/ 20 in 2 quoins est (4-J 5." means 8-1/20, divided by 2,
gives the quotient 4— j/5- Similarly,6 "Ir. /1 12 — /76" meant
1/1/112 — 1/76; the r signifying here residua, or "remainder," and
therefore Ir. signified the square root of the binomial difference.
In the 1592 edition7 of Ramus' arithmetic and algebra, edited by
Lazarus Schoner, "Ic 4" stands for (^4, and "I bq 5" for 1/5, in place of
Ramus' "II 5." Also, 1/2. 1/3 = 1/6, 1/6 -v- 1/2 = 1/3 is expressed
thus:8
"Esto multiplicandum 12 per J3 factus erit I 6.
12. I6f „
13. /2V3'
16.
It is to be noted that with Schoner the I received an extension of
meaning, so that 51 and Z5, respectively, represent 5x and 1/5, the I
standing for the first power of the unknown quantity when it is not
1 Die Schriften der romischen Feldmesser (ed. Blume, Lachmann, Rudorff;
Berlin, 1848-52), Vol. I, p. 96.
2 Martianus Capella, DeNuptiis (ed. Kopp; Frankfort, 1836), lib. VII, § 748.
3 Gerberti opera mathematica (ed. Bubnow; Berlin, 1899), p. 83. See J. Tropfke,
op. cit., Vol. II (2d ed., 1921), p. 143.
4 P. Rami Scholarvm mattiematicarvm libri unus et triginti (Basel, 1569),
Lib. XXIV, p. 276, 277.
6 Ibid., p. 179.
*Ibid., p. 283,
7 Petri Rami ... Arithmeticea libri duo, et algebrae totidem: a Lazaro Schoner o
(Frankfurt, 1592), p. 272 ff.
8 Petri Rami ... Arithmetical libri duo et geometriae septem et vigintit Dudum
quidem, a Lazaro Schonero (Frankfurt a/M., 1627), part entitled "De Nvmeri
figvratis Lazari Schoneri liber/ ' p. 178.
ROOTS 365
followed by a number (see also § 290). A similar change in meaning
resulting from reversing the order of two symbols has been observed
in Pacioli in connection with 5 (§§ 136, 137) and in A. Girard in
connection with the circle of Stevin (§ 164). The double use of the
sign I, as found in Schoner, is explained more fully by another pupil of
Rarnus, namely, Bernardus Salignacus (§ 291).
Ramus' I was sometimes used by the great French algebraist
Francis Victa who seemed disinclined to adopt either R or ]/ for
indicating roots (§ 177).
This use of the letter I in the calculus of radicals never became
popular. After the invention of logarithms, this letter was needed to
mark logarithms. For that reason it is especially curious that Henry
Briggs, who devoted the latter part of his life to the computation and
the algorithm of logarithms, should have employed I in the sense as-
signed it by llamus and Vieta. In 1624 Briggs used /, i(3), II for square,
cube, and fourth root, respectively. "Sic /(3) 8 [i.e., i? 8], latus cubicum
Octonarii, id est 2. sic / bin 2 -f 13. [i.e., ^ 2+1/3] latus binomii
2+Z3." Again, "II 85 J [i.e., ^8511. Latus 85 J est O**-™*4-30-*, et
huius lateris latus est S03--51-3-43"2-4-. cui numero aequatur // 85J."1
323. Napier1 s line symbolism. — John Napier2 prepared a manu-
script on algebra which was not printed until 1839. He made use of
StifcFs notation for radicals, but at the same time devised a new
scheme of his own. "It is interesting to notice that although Napier
invented an excellent notation of his own for expressing roots, he did
not make use of it in his algebra, but retained the cumbrous, and in
some cases ambiguous notation generally used in his day. His nota-
tion was derived from this figure
1 |2
4 | 5
71 8
3
9
in the following way: U prefixed to a number means its square root,
ID its fourth root, D its fifth root, T its ninth root, and so on, with
extensions of obvious kinds for higher roots. "3
1 Henry Briggs, Arithmetica logarithmica (London, 1624), Introduction.
2 De Arte Logistica Joannis Naperi Merchistonii Baronis Libri qui super aunt
(Edinburgh, 1839), p. 84.
3 J. E. A. Steggall, "De arte logistica," Napier Tercentenary Memorial Volume
(ed. Cargill Gilston Knott; London, 1915), p. 160.
366 A HISTORY OF MATHEMATICAL NOTATIONS
THE SIGN l/
324. Origin of i/. — This symbol originated in Germany. L. Euler
guessed that it was a deformed letter r, the first letter in radix.1 This
opinion was held generally until recently. The more careful study of
German manuscript algebras and the first printed algebras has con-
vinced Germans that the old explanation is hardly tenable; they have
accepted the a priori much less probable explanation of the evolution
of the symbol from a dot. Four manuscript algebras have been avail-
able for the study of this and other questions.
The oldest of these is in the Dresden Library, in a volume of manu-
scripts which contains different algebraic treatises in Latin and one
in German.2 In one of the Latin manuscripts (see Fig. 104, A 7),
probably written about 1480, dots are used to signify root extraction.
In one place it says: "In extraccione radicis quadrati alicuius numeri
preponatur nurnero vnus punctus. In extraccione radicis quadrati
radicis quadrati prepone numero duo puncta. In extraccione cubici
radicis alicuius numeri prepone tria puncta. In extraccione cubici
radicis alicuius radicis cubici prepone 4 puncta."3 That is, one dot (•)
placed before the radicand signifies square root; two dots (..) signify
the square root of the square root; three dots (...) signify cube root;
four dots (....), the cube root of the cube root or the ninth root. Evi-
dently this notation is not a happy choice. If one dot meant square
root and two dots meant square root of square root (i.e., 1/j/ ), then
three dots should mean square root of square root of square root, or
eighth root. But such was not actually the case; the three dots
were made to mean cube root, and four dots the ninth root. What was
the origin of this dot-system? No satisfactory explanation has been
found. It is important to note that this Dresden manuscript was once
in the possession of Joh. Widman, and that Adam Riese, who in 1524
prepared a manuscript algebra of his own, closely followed the
Dresden algebra.
325. The second document is the Vienna MS4 No. 5277, Ilegule-
1 L. Euler, Institutiones calculi differentialis (1775), p. 103, art. 119; J. Tropfke,
07?. cit., Vol. II (2d ed., 1921), p. 150. ,
2 M. Cantor, Varies, uber Geschichte der Malhematik, Vol. II (2. Aufl., 1900),
p. 241.
3E. Wappler, Zur Geschichte der deulschen Algebra im XV. Jahrhundert,
Zwickauer Gyrnnasialprogramm von 1887, p. 13. Quoted by J. Tropfke, op. tit.,
Vol. II (1921), p. 146, and by M. Cantor, op. tit., Vol. II (2. Aufl., 1900), p. 243.
4C. J. Gerhardt, Monatsberichte Akad. (Berlin, 1867), p. 46; ibid. (1870),
p. 143-47; Cantor, op. cit., Vol. II (2d ed., 1913), p. 240, 424.
ROOTS 367
Cose-uel Algobre-. It contains the passage: "Quum 3 assimiletur
radici de radice punctus deleatur de radice, 3 in se ducatur et remanet
adhuc inter se aequalia"; that is, "When x2 = V/x, erase the point
before the x and multiply x2 by itself, then things equal to each other
are obtained." In another place one finds the statement, per punctum
intellige radicem — "by a point understand a root." But no dot is
actually used in the manuscript for the designation of a root.
The third manuscript is at the University of Gottingen, Codex
Gotting. Philos. 30. It is a letter written in Latin by Initius Algebras,1
probably before 1524. An elaboration of this manuscript was made
in German by Andreas Alexander.2 In it the radical sign is a heavy
point with a stroke of the pen up and bending to the right, thus /.
It is followed by a symbol indicating the index of the root; /$ indi-
cates square root; /ce, cube root; /cc% the ninth root, etc. More-
over, /cs|8+/22j stands for * 8+1/22, where cs (i.e., communis)
signifies the root of the binomial which is designated as one quantity,
by lines, vertical and horizontal. Such lines are found earlier in
Chuquet (§ 130). The $, indicating the square root of the binomial,
is placed as a subscript after the binomial. Calling these two lines a
"gnomon," M\ Curtze adds the following:
"This gnomon has here the signification, that what it embraces is
not a length, but a power. Thus, the simple 8 is a length or simple
number, while [83 is a square consisting of eight areal units whose
linear unit is /$|8. In the same way [8C* would be a cube, made up of
8 cubical units, of which fce 8 is its side, etc. A double point, with the
tail attached to the last, signifies always the root of the root. For
example, ,/cc[88 would mean the cube root of the cube root of 88. It
is identical with /cc'88, but is used only when the radicand is a
so-called median [Mediale] in the Euclidean sense."3
326. The fourth manuscript is an algebra or Coss completed by
Adam Riese4 in 1524; it was not printed until 1892. Riese was familiar
with the small Latin algebra in the Dresden collection, cited above;
1 Initius Algebras: Algebrae Arabis Arithmetici viri clarrisimi Liber ad Ylem
geometram magislrum suum. This was published by M. Curtze in Abhandlungen
zur Geschichte der mathematischen Wissenschaften, Heft XIII (1902), p. 435-611.
Matters of notation are explained by Curtze in his introduction, p. 443-48.
2 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. Ill (1902), p. 355-60.
3 M. Curtze, op. cit.t p. 444,
4 B. Berlet, Adam Riese, sein Leben, seine Rechenbucher und seine Art zu
rechnen; die Coss von Adam Riese (Leipzig-Frankfurt a/M., 1892).
368 A HISTORY OF MATHEMATICAL NOTATIONS
he refers also to Andreas Alexander.1 For indicating a root, Riese
does not use the dot, pure and simple, but the dot with a stroke
attached to it, though the word punct ("point ") occurs. Riese says:
"1st, so $ vergleicht wird j/ vom radix, so mal den g in sich multipli-
ciren vnnd das punct vor dem Radix aussleschn."2 This passage has
the same interpretation as the Latin passage which we quoted from
the Vienna manuscript.
We have now presented the main facts found in the four manu-
scripts. They show conclusively that the dot was associated as a sym-
bol with root extraction. In the first manuscript, the dot actually
appears as a sign for roots. The dot does not appear as a sign in the
second manuscript, but is mentioned in the text. In the third and
fourth manuscripts, the dot, pure and simple, does not occur for the
designation of roots; the symbol is described by recent writers as a
dot with a stroke or tail attached to it. The question arises whether
our algebraic sign j/ took its origin in the dot. Recent German writers
favor that view, but the evidence is far from conclusive. Johannes
Widman, the author of the Rechnung of 1489, was familiar with the
first manuscript which we cited. Nevertheless he does not employ
the dot to designate root, easy as the symbol is for the printer. He
writes clown $ and ra. Christoff Rudolff was familiar with the Vienna
manuscript which uses the dot with a tail. In his Coss of 1525 he
speaks of the Punkt in connection with root symbolism, but uses a
mark with a very short heavy downward stroke (almost a point),
followed by a straight line or stroke, slanting upward (see Fig. 59). As
late as 1551, Scheubel,3 in his printed Algebra, speaks of points. He
says: "Solent tamen multi, et bene etiam, has desideratas radices,
suis punctis cu linea quadam a dextro latere ascendente, notare "
("Many are accustomed, and quite appropriately, to designate the
desired roots by points, from the right side of which there ascends a
kind of stroke/1) It is possible that this use of "point" was technical,
signifying "sign for root," just as at a later period the expression
"decimal point" was used even when the symbol actually written
down to mark a decimal fraction was a comma. It should be added
that if Rudolff looked upon his radical sign as really a dot, he would
have been less likely to have used the dot again for a second purpose
in his radical symbolism, namely, for the purpose of designating that
1 B. Berlet, op. tit., p. 29, 33.
2 C. I. Gerhardt, op. cit. (1870) p. 151.
3J. Scheubel, Algelvra compendiosa (Paris, 1551), fol. 25B. Quoted from J.
Tropfke, op. cil.t Vol. II (2d ed., 1921), p. 149.
ROOTS 369
the root extraction must be applied to two or more terms following the
I/; this use of the dot is shown in § 148. It is possible, perhaps prob-
able, that the symbol in Rudolff and in the third and fourth manu-
scripts above referred to is not a point at all, but an r, the first letter
in radix. That such was the understanding of the sixteenth-century
Spanish writer, Perez de Moya (§ 204), is evident from his designa-
tions of the square root by r, the fourth root by rr, and the cube root
by rrr. It is the notation found in the first manuscript which we cited,
except that in Moya the r takes the place of the dot ; it is the notation
of Rudolff, except that the sign in Rudolff is not a regularly shaped r.
In this connection a remark of H. Wieleitner is pertinent: "The dot
appears at times in manuscripts as an abbreviation for the syllable
ra. Whether the dot used in the Dresden manuscript represents this
normal abbreviation for radix does not appear to have been specially
examined. J)l
The history of our radical sign j/, after the time of Rudolff, relates
mainly to the symbolisms for indicating (1) the index of the root,
(2) the aggregation of terms when the root of a binomial or polynomial
is required. It took over a century to reach some sort of agreement
on these points. The signs of Christoff Rudolff are explained more
fully in § 148. Stifel's elaboration of the symbolism of Andreas Alex-
ander as given in 1544 is found in §§ 153, 155. Moreover, he gave to
the y its modern form by making the heavy left-hand, downward
stroke, longer than did Rudolff.
327. Spread of y. — The German symbol of y for root found its
way into France in 1551 through Scheubers publication (§159);
it found its way into Italy in 1608 through Clavius; it found its way
into England through Recorde in 1557 (§ 168) and Dee in 1570
(§ 169); it found its way into Spain in 1552 through Marco Aurel
(§§ 165, 204), but in later Spanish texts of that century it was super-
seded by the Italian /?. The German sign met a check in the early
works of Vieta who favored Ramus' Z, but in later editions of Vieta,
brought out under the editorship of Fr. van Schooten, the sign y
displaced Vieta's earlier notations (§ 176, 177).
In Denmark Chris. Dibuadius2 in 1605 gives three designations
of square root, y7, i/Q, j/g; also three designations of cube root,
1/C, j/c, yce] and three designations of the fourth root, j/j/, i/QO,
1/JJ.
Stifel's mode of indicating the order of roots met with greater
1 H. Wieleitner, Die Sicben Rechnungsarten (Leipzig-Berlin, 1912), p. 49.
2 C. Dibvadii in Arithmeticam irrationalivm Euclidis (Arnhem, 1005).
370 A HISTORY OF MATHEMATICAL NOTATIONS
general favor than Rudolff's older and clumsier designation (§§ 153,
155).
328. Rudolff's signs outside of Germany. — The clumsy signs of
Christoff Rudolff, in place of which Stifel had introduced in 1544 and
1553 better symbols of his own, found adoption in somewhat modified
form among a few writers of later date. They occur in Aurel's Spanish
Arithmeticat 1552 (§ 165). They are given in Recorde, Whetstone of
Witte (1557) (§ 168), who, after introducing the first sign, j/., pro-
ceeds: "The seconde signe is annexed with Surde Cubes, to expresse
their rootes. As this ./w\/«16 whiche signifieth the Cubike roote of .16.
And ./VW/.20. betokeneth the Cubike roote of .20. Andsoforthe. But
many tymes it hath the Cossike signe with it also: as /wv/-c* 25 the
Cubike roote of .25. And /wv/-c*.32. the Cubike roote of .32. The
thirde figure doeth represente a zenzizenzike roote. As ./\\/.l2. is the
zenzizenzike roote of .12. And /vs/.35. is the zenzizenzike roote of .35.
And likewaies if it haue with it the Cossike signe .gj. As A\/3324 the
zenzizenzike roote of .24. and so of other."
The Swiss Ardiiser in 1627 employed Rudolff's signs for square
root and cube root.1 J. H. Rahn in 1659 used /vv\/ for evolution,2
which may be a modified symbol of Rudolff; Rahn's sign is adopted
by Thomas Brancker in his English translation of Rahn in 1668, also
by Edward Hatton3 in 1721, and by John Kirkby4 in 1725. Ozanam5
in 1702 writes V//5+ v'2 and also M/5+WA/2. Samuel Jeake6 in 1696
gives modifications of Rudolff's signs, along with other signs, in an
elaborate explanation of the "characters" of "Surdes"; j/ means
root, j/: or V or VV universal root, AA/ or Vl square root, AW/ or
]/</> cube root, /vwx/ or j/jj squared square root, /wwv/ or j/p
sursolide root.
On the Continent, Johann Caramuel7 in 1670 used j/ for square
root and repeated the symbol i/j/ for cube root: "i/j/27. est Radix
Cubica Numeri 27. hoc est, 3."
1 Johann Ardiiser, Geometriae Thearicae et Practicae, XII. Bucher (Zurich,
1627), fol. 8L4.
2 Johann Heinrich Rahn, Teuische Algebra (Zurich, 1659).
8 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 287.
4 John Kirkby, Arithmetical Institutions (London, 1735), p. 7.
6 J. Ozanam, Nouveaux Siemens d'algebre ... par M. Ozanam, I. Par tie (Am-
sterdam, 1702), p. 82.
8 Samuel Jeake, AOHSTIKHAOrf A, or Arithmetick (London, 1696), p. 293.
1 Joannis Caramvdis Mathesis Biceps. Veiu8> et Nova (Campaniae, 1670),
p. 132.
ROOTS 371
329. Stevin's numeral root-indices. — An innovation of considerable
moment were Stevin's numeral indices which took the place of Stifel's
letters to mark the orders of the roots. Beginning with Stifel the
sign ]/ without any additional mark came to be interpreted as mean-
ing specially square root. Stevin adopted this interpretation, but in
the case of cube root he placed after the j/ the numeral 3 inclosed in
a circle (§§ 162, 163). Similarly for roots of higher order. Stevin's
use of numerals met with general but not universal adoption. Among
those still indicating the order of a root by the use of letters was Des-
cartes who in 1637 indicated cube root by j/C. But in a letter of 1640
he1 used the 3 and, in fact, leaned toward one of Albert Gjrard's
notations, when he wrote |/3).20+|/392 for ^20+1/392. But
very great diversity prevailed for a century as to the exact position
of the numeral relative to the |/. Stevin's j/, followed by numeral
indices placed within circles, was adopted by Stampioen,2 and by
van Schooten.3
A. Romanus displaced the circle of Stevin by two round parenthe-
ses, a procedure explained in England by Richard Sault4 who gives
or a+b\*. Like Girard, Harriot writes j/3.)26+i/675 for
^26 +1/675 (see Fig. 87 in § 188). Substantially this notation was
used by Descartes in a letter to Mersenne (September 30, 1640),
where he represents the ratine cubique by i/3), the ratine sursolide by
1/5), the B sursolide by j/7), and so on.5 Oughtred sometimes used
square brackets, thus v/[12]1000 for I^IOOO (§ 183).
330. A step in the right direction is taken by John Wallis6 who in
1655 expresses the root indices in numerals without inclosing them in
a circle as did Stevin, or in parentheses as did Romanus. However,
Wallis' placing of them is still different from the modern; he writes
j/3/22 for our f/R2. The placing of the index within the opening of the
radical sign had been suggested by Albert Girard as early as 1629.
Wallis' notation is found in the universal arithmetic of the Spaniard,
Joseph Zaragoza,7 who writes j/4243 - y*%j for our 1^243- ^27, and
1 (Euvres de Descartes, Vol. X, p. 190.
2 Algebra ofte Nieuwe Stel-Regel ... door Johan Stampioen d'Jongle 's Graven-
Hage (1639), p. 11.
3 Fr. van Schooten, Geometria a Renato des Cartes (1649), p. 328.
4 Richard Sault, A New Treatise of Algebra (London, n.d.).
6 (Euvres de Descartes, Vol. Ill (1899), p. 188.
6 John Wallis, Arilhmetica infinitorum (Oxford, 1655), p. 59, 87, 88.
7 Joseph Zaragoza, Arithmetica universal (Valencia, 1669), p. 307.
372 A HISTORY OF MATHEMATICAL NOTATIONS
l/2(7+i/213) for our ^7 +1/13. Wallis employs this notation1
again in his Algebra of 1685. It was he who first used general indices2
in the expression i/dRd — R. The notation i/4 19 for 1/19 crops out
again3 in 1697 in De Lagny's j/354 — 1/316 = j/32; it is employed by
Thomas Walter;4 it is found in the Maandelykse Mathematische Lief-
hebberye (1754-69), though the modern \/ is more frequent; it is given
in Castillion's edition5 of Newton's Arithmetica universalis.
331. The Girard plan of placing the index in the opening of the
radical appears in M. llolle's Traite d'Algebre (Paris, 1690), in a letter
of Leibniz6 to Varignon of the year 1702, in the expression 1/1 + j/ — 3,
and in 1708 in (a review of) G. Manfred7 with literal index, *\aa+bbn.
At this time the Leibnizian preference for i/(aa+bb) in place of
l/aa+66 is made public;8 a preference which was heeded in Germany
and Switzerland more than in England and France. In Sir Isaac
Newton's Arithmetica universalis9 of 1707 (written by Newton some-
time between 1673 and 1683, and published by Whiston without hav-
ing secured the consent of Newton) the index numeral is placed after
the radical, and low, as in |/3 : 64 for 1/64, so that the danger of con-
fusion was greater than in most other notations.
During the eighteenth century the placing of the root index in
the opening of the radical sign gradually came in vogue. In 1732 one
finds 1/25 in De la Loubere;10 De Lagny11 who in 1697 wrote j/3, in
1733 wrote ^/~\ Christian Wolff12 in 1716 uses in one place the astro-
1 John Wallis, A Treatise of Algebra (London, 1685), p. 107; Opera, Vol. II
(1693), p. 118. But see also Arithmetica infmitorum (1656), Prop. 74.
2 Mathesis universalis (1657), p. 292.
3 T. F. de Lagny, Nouveaux elemens d'arithmetique el d'algcbre (Paris, 1697),
p. 333.
4 Thomas Walter, A new Mathematical Dictionary (London, n.d., but pub-
lished in 1762 or soon after), art. "Heterogeneous Surds."
5 Arithmetica universalis .... auclore Is. Newton .... cum commenlario
Johannis Castillionei . . . . , Tomus primus (Amsterdam, 1761), p. 76.
0 Journal des S^avans, anne"e 1702 (Amsterdam, 1703), p. 300.
7 Ibid., ann6e 1708, p. 271. 8 Ibid.
9 Isaac Newton, Arithmetica universalis (London, 1707), p. 9; Tropfke, Vol.
II, p. 154.
10 Simon de la Loubere, De la Resolution des Equations (Paris, 1732), p. 119.
11 De Lagny in Memoir es de Vacademie r, des sciences, Tome XI (Paris, 1733),
p. 4.
12 Christian Wolff, Mathematisches Lexicon (Leipzig, 1716), p. 1081.
ROOTS 373
nomical character representing Aries or the ram, for the radical sign,
and writes the index of the root to the right; thus T3 signifies cube
root. Edward Hatton1 in 1721 uses f, \/, ty\ De la Chapelle2 in 1750
wrote tyW. Wolff3 in 1716 and Hindenburg4 in 1779 placed the index
to the left of the radical sign, 3]/Z; nevertheless, the notation \/
came to be adopted almost universally during the eighteenth century.
Variations appear here and there. According to W. J, Greenstrcet,6
a curious use of the radical sign is to be found in Walkingame's
Tutor's Assistant (20th ed., 1784). He employs the letter V for square
root, but lets F3 signify cube or third power, F4 the fourth power. On
the use of capital letters for mathematical signs, very often encountered
in old books, as F, for j/, > for >, Greenstreet remarks that "authors
in the eighteenth century complained of the meanness of the Cam-
bridge University Press for using daggers set sideways instead of the
usual + •" In 1811, an anonymous arithmetician6 of Massachusetts
suggests 24 for 1//4, 38 for ^8, m8 for l/'S.
As late as 1847 one finds7 the notaton 3|/fe, ml/a&c, for the cube
root and the wth root, the index appearing in front of the radical sign.
This form was not adopted on account of the limitations of the print-
ing office, for in an article in the same series, from the pen of De
Morgan, the index is placed inside the opening of the radical sign.8
In fact, the latter notation occurs also toward the end of Parker's
book (p. 131).
In a new algorithm in logarithmic theory A. Biirja9 proposed the
sign i/ a to mark the nth root of the order N, of a, or the number of
which the nth power of the order N is a.
1 Edward Hatton, op. cit. (London, 1721), p. 287.
2 De la Chapelle, Traite des sections coniques (Paris, 1750), p. 15.
3 Christian Wolff, Mathematisches Lexicon (Leipzig, 1716), "Signa," p. 1265.
4 Carl F. Hindenburg, Infmitinomii dignitatum leges ac formulae (Gottingcn,
1779), p. 41.
5 W. J. Greenstreet in Mathematical Gazette, Vol. XI (1823), p. 315.
6 The Columbian Arithmetician, "by an American" (Havershall, Mass.,
1811), p. 13.
7 Parker, "Arithmetic and Algebra," Library of Useful Knowledge (London,
1847), p. 57.
8 A. de Morgan, "Study and Difficulties of Mathematics," ibid., Mathematics,
Vol. I (London, 1847), p. 56.
9 A. Biirja in Nouveaux memoires d. I' academic r. d. scienc. et bett.-lett.t anne"e
1778 et 1779 (Berlin, 1793), p. 322.
374 A HISTORY OF MATHEMATICAL NOTATIONS
332. Rudolff and Stifel's aggregation signs. — Their dot symbolism
for the aggregation of terms following the radical sign •]/ was used bv
Peletier in 1554 (§ 172). In Denmark, Chris. Dibuadius1 in 1605
marks aggregation by one dot or two dots, as the case may demand.
Thus v/.S+j/S. +1/2 means 1/5+1/3+1/2; i/.5+|/3+i/2means
1/3 + 1/2 .
W. Snell's translation2 into Latin of Ludolf van Ceulen's book on
the circle contains the expression
which is certainly neater than the modern
V
2
The Swiss, Johann Ardtiser,3 in 1627, represents i/(2 — j/3) by
V.2-*V3" and i/[2+i/(2+l/2)] by V-2+i/-2+i/2." This
notation appears also in one of the manuscripts of Reri6 Descartes,4
written before the publication of his Geomitrie in 1637.
It is well known that Oughtred in England modified the German
dot symbolism by introducing the colon in its place (§ 181). He had
settled upon the dot for the expression of ratio, hence was driven to
alter the German notation for aggregation. Oughtred's5 double
colons appear as in "i/q:aq — eq:" for our j/(a2 — e2).
We have noticed the use of the colon to express aggregation, in
the manner of Oughtred, in the Arithmetique made easie, by Edmund
Wingate (2d ed. by John Kersey; London, 1650), page 387; in John
Wallis' Operum mathematicorum pars altera (Oxonii, 1656), page 186,
as well as in the various parts of Wallis' Treatise of Algebra (London,
1685) (§ 196), and also in Jonas Moore's Arithmetick in two Books
(London, 1660), Second Part, page 14. The 1630 edition of Wingate's
book does not contain the part on algebra, nor the symbolism in
question; these were probably added by John Kersey.
1 C. Dibvadii in arithmeticam irrationalivm Evdidis (Arnhem, 1605), Intro-
duction.
2 Lvdolphi d Cevlen de Circvlo Adscriptis Liber ... omnia 6 vernaculo Latina
fecit ... Willebrordus Sndlius (Ley den, 1610), p. 5.
3 Johann Ardliser, Geometriae, Theweticae practicae, XII. Biicher (Zurich,
1627), p. 97, 98.
4 (Euvres de Descartes, Vol. X (1908) p. 248.
8 Eudidis dedaratio, p. 9, in Oughtred's Ctavis (1652).
ROOTS 375
333. Descartes9 union of radical sign and vinculum. — Ren6 Des-
cartes, in his Geometric (1637), indicates the cube root by j/C. as in
for our
Here a noteworthy innovation is the union of the radical sign }/ with
the vinculum (§ 191). This union was adopted in 1640 by J. J.
Stampioen,1 but only as a redundant symbol. It is found in Fr. van
Schooten's 1646 edition of the collected works of Vieta (§ 177), in
van Schooten's conic sections,2 as also in van Schooten's Latin edi-
tion of Descartes' geometry.8 It occurs in J. H. Rahn's algebra (1659)
and in Brancher's translation of 1668 (§ 194).
This combination of radical sign -[/ and vinculum is one which has
met with great favor and has maintained a conspicuous place in
mathematical books down to our own time. Before 1637, this combi-
nation of radical sign and vinculum had been suggested by Descartes
(CEuvres, Vol. X, p. 292). Descartes also leaned once toward Girard's
notation.
Great as were Descartes' services toward perfecting algebraic
notation, he missed a splendid opportunity of rendering a still greater
service. Before him Oresme and Stevin had advanced the concept of
fractional as well as of integral exponents. If Descartes, instead of
extending the application of the radical sign j/ by adding to it the
vinculum, had discarded the radical sign altogether and had intro-
duced the notation for fractional as well as integral exponents, then
it is conceivable that the further use of radical signs would have been
discouraged and checked; it is conceivable that the unnecessary dupli-
cation in notation, as illustrated by 6* and 1/63, would have been
avoided; it is conceivable that generations upon generations of pupils
would have been saved the necessity of mastering the operations with
two difficult notations when one alone (the exponential) would have
answered all purposes. But Descartes missed this opportunity, as
did later also I. Newton who introduced the notation of the fractional
exponent, yet retained and used radicals.
1 J. J. Stampioen, Wis-Konstich ende Reden-Maetich Bewijs (The Hague, 1640),
p. 6.
2 Francisd & Schooten Leydensis de organica conicarum sectionum (Leyden,
1646), p. 91.
3 Francisci a Schooten, Renati Descartes Geomelria (Frankfurt a/M., 1695),
p. 3. [First edition, 1649.]
376 A HISTORY OF MATHEMATICAL NOTATIONS
334. Other signs of aggregation of terms. — Leonard and Thomas
Digges,1 in a work of 1571, state that if "the side of the Pentagon, [is]
14, the containing circles scmidiameter [is]
V/S.F.98+i/sl920f ' i.e., VQS+ i/1920| .
In the edition of 159 12 the area of such a pentagon is given as
1/3 mi. 60025+1/32882400500 .
Vieta's peculiar notations for radicals of 1593 and 1595 are given
in § 177. The Algebra of Herman Follinvs3 of 1622 uses parentheses in
connection with the radical sign, as in V/g(22+V/g9), our 1/22 +1/9.
Similarly, Albert Girard4 writes i/(2J + l/3-J), with the simplifica-
tion of omitting in case of square root the letter marking the order of
the root. But, as already noted, he does not confine himself to this
notation. In one place5 he suggests the modern designation i/7, f/, i/.
Oughtred writes i/u or j/b for universal root (§ 183), but more
commonly follows the colon notation (§ 181). Herigone's notation of
1634 and 1644 is given in § 189. The Scotsman, James Gregory,6
writes
J5 1fi22\
In William Molyneux7 one finds VCP*-Px* for Vpi-Px*. Another
mode of marking the root of a binomial is seen in a paper of James
Bernoulli8 who writes >/, ax— x2 for Vax — x2. This is really the old
idea of Stifel, with Herigone's and Leibniz's comma taking the place
of a dot.
The union of the radical sign and vinculum has maintained itself
widely, even though it had been discouraged by Leibniz and others
who aimed to simplify the printing by using, as far as possible, one-
line symbols. In 1915 the Council of the London Mathematical So-
1 A Geometrical Practise, named Pantometria, framed by Leonard Digges t ....
finished by Thomas Digges his sonne (London, 1571) (pages unnumbered).
2 A Geometrical Practical Treatize named Pantometria (London, 1591), p. 106.
3 Hermann Follinvs, Algebra sive liber de Rebvs occvltis (Cologne, 1622), p. 157.
4 Albert Girard, Invention nouvelle en ljalgebre (Amsterdam, 1629).
5 Loc. cit.y in "Caracteres de puissances et racines."
6 James Gregory, Geometriae pars vniversalis (Patavii, 1668), p. 71, 108.
7 William Molyneux, A Treatise of Dioptricks (London, 1692), p. 299.
8 Jacob Bernoulli in Ada eruditorum (1697), p. 209.
ROOTS 377
ciety, in its Suggestions for Notation and Printing ,l recommended that
1/2 or 2* be adopted in place of 1/2, also i/(ax2+2bx+c) or (ax2+
in place of V/ax2+2bx+c. Bryan2 would write ]/— 1 rather
than 1/^T.
335. Redundancy in the use of aggregation signs. — J. J. Stampioen
marked aggregation of terms in three ways, any one of which would
have been sufficient. Thus,3 he indicates l/63+6a262+9«46 in this
manner, l/.(666+6aa 66+9aaaa 6); he used here the vinculum, the
round parenthesis, and the dot to designate the aggregation of the
three terms. In other places, he restricts himself to the use of dots,
either a dot at the beginning and a dot at the end of the expression,
or a dot at the beginning and a comma at the end, or he uses a dot
and parentheses.
Another curious notation, indicating fright lest the aggregation
of terms be overlooked by the reader, is found in John Kersey's
symbolism of 1673,4 1/(2):Jr-l/irr-a: for \r-V\r*-&. We
observe here the superposition of two notations for aggregation, the
Oughtredian colon placed before and after the binomial, and the
vinculum. Either of these without the other would have been suffi-
cient.
336. Peculiar Dutch symbolism. — A curious use of |/ sprang up in
Holland in the latter part of the seventeenth century and maintained
itself there in a group of writers until the latter part of the eighteenth
century. If ;/ is placed before a number it means "square root/' if
placed after it means "square." Thus, Abraham de Graaf5 in 1694
indicates by \/-^r ^e square root of the fraction, by n~\ / the square
\ 0*2 JJ5 \
of the fraction. This notation is used often in the mathematical jour-
nal, Maandelykse Mathematische Liefhebberye, published at Amster-
dam from 1754 to 1769. As late as 1777 it is given by L. Praalder8 of
Utrecht, and even later (1783) by Pieter Venema.7 We have here the
1 Mathematical Gazette, Vol. VIII (1917), p. 172.
2 Op. cit., Vol. VIII, p. 220.
3 J. J. Stampioenii Wis-Konstigh Ende Reden-Maetigh Bewjs (The Hague,
1640), p. 6.
4 John Kersey, Algebra (London, 1673), p. 95.
6 Abraham de Graaf, De Geheele Mathesis (Amsterdam, 1694), p. 65, 69.
6 Laurens Praalder, Mathematische Voorstellen (Amsterdam, 1777), p. 14, 15 ff .
7 Pieter Venema, Algebra ofte Stel-Konst, Vyfde Druk (Amsterdam, 1783),
p. 168, 173.
378 A HISTORY OF MATHEMATICAL NOTATIONS
same general idea that was introduced into other symbolisms, accord-
ing to which the significance of the symbol depends upon its relative
position to the number or algebraic expression affected. Thus with
Pacioli 5200 meant 1/200, but #3° meant the second power (§§ 135,
136). With Stevin (§ 162, 163), ®20 meant 203, but 200 meant
20s3. With L. Schoner 5J meant 5z, but 15 meant 1/5 (§ 291). We
may add that in the 1730 edition of Venema's algebra brought out in
New York City radical expressions do not occur, as I am informed by
Professor L. G. Simons, but a letter placed on the left of an equation
means division of the members of the equation by it; when placed
on the right, multiplication is meant. Thus (p. 100) :
«
6
and (p. 112):
"5_4500 . 1000
x x^
5:^500 = Hz -1000
Similar is PrandePs use of -j/ as a marginal symbol, indicating
that the square root of both sides of an equation is to be taken. His
marginal symbols are shown in the following:1
I/
337. Principal root-values. — For the purpose of distinguishing be-
tween the principal value of a radical expression and the other values,
G. Peano2 indicated by i/*a all the m values of the radical, reserving
i/a for the designation of its "principal value." This notation is
adopted by O. Stolz and J. A. Gmeiner3 in their Theoretische Arith-
metik (see also § 312).
1 J. G. Prandel, Kugldreyeckslehre und hohere Mathematik (Miinchen, 1793),
p. 97.
2 G. Peano, Formidaire des math&natiques (first published in Rivista di Mate-
matica ), Vol. I, p. 19.
3 0. Stolz und J. A. Gmeiner, Thevrelische Arithmetik (Leipzig), Vol. II (1902),
p. 355.
UNKNOWN NUMBERS 379
338. Recommendation of United States National Committee. —
"With respect to the root sign, i/, the committee recognizes that
convenience of writing assures its continued use in many cases instead
of the fractional exponent. It is recommended, however, that in
algebraic work involving complicated cases the fractional exponent
be preferred. Attention is called to the fact that the symbol I/a
(a representing a positive number) means only the positive square
root and that the symbol i^a means only the principal nth root, and
i^
similarly for a*, a71."1
SIGNS FOR UNKNOWN NUMBERS
339. Early forms. — Much has already been said on symbolisms
used to represent numbers that are initially unknown in a problem, and
which the algebraist endeavors to ascertain. In the Ahmes papyrus
there are signs to indicate "heap" (§23); in Diophantus a Greek
letter with an accent appears (§ 101); the Chinese had a positional
mode of indicating one or more unknowns; in the Hindu Bakhshali
manuscript the use of a dot is invoked (§ 109). Brahmagupta and
Bhaskara did not confine the symbolism for the unknown to a single
sign, but used the names of colors to designate different unknowns
(§§ 106, 108, 112, 114). The Arab Abu Kamil2 (about 900 A.D.), modi-
fying the Hindu practice of using the names of colors, designated the
unknowns by different coins, while later al-Karkhi (following perhaps
Greek sources)3 called one unknown "thing," a second "measure" or
"part," but had no contracted sign for them. Later still al-Qalasddi
used a sign for unknown (§ 124). An early European sign is found in
Regiomontanus (§ 126), later European signs occur in Pacioli (§§ 134,
136), in Christoff Rudolff (§§ 148, 149, 151), 4 in Michael Stifel who
used more than one notation (§§ 151, 152), in Simon Stevin (§ 162), in
L. Schoner (§ 322), in F. Vieta (§§ 176-78), and in other writers
(§§ 117, 138, 140, 148, 164, 173, 175, 176, 190, 198).
Luca Pacioli remarks5 that the older textbooks usually speak of
1 Report of the National Committee on Mathematical Requirements under the
Ausjyices of the Mathematical Association of America (1923), p. 81.
2 H. Suter, Bibliotheca mathematics (3d ser.), Vol. XI (1910-11), p. 100 ff.
3 F. Woepcke, Extrait du Fakhri (Paris, 1853), p. 3, 12, 13&-43. See M.
Cantor, op. cit., Vol. I (3d cd., 1907), p. 773.
4 Q. Enestrom, Bibliotheca mathematics (3d ser.), Vol. VIII (1907-8), p. 207.
5L. Pacioli, Summa, dist. VIII, tract 6, fol. 148#. See M. Cantor, op. cit.,
Vol. II (2d ed., 1913), p. 322.
380 A HISTORY OF MATHEMATICAL NOTATIONS
the first and the second cosa for the unknowns, that the newer writers
prefer cosa for the unknown, and quantita for the others. Pacioli
abbreviates those co. and #K
Vieta's convention of letting vowels stand for unknowns and
consonants for knowns (§§ 164, 176) was favored by Albert Girard,
and also by W. Oughtred in parts of his Algebra, but not throughout.
Near the beginning Oughtred used Q for the unknown (§182).
The use of N (numerus) for x in the treatment of numerical equa-
tions, and of Q, C, etc., for the second and third powers of x, is found
in Xylander's edition of Diophantus of 1575 (§ 101), in Vieta's De emen-
datione aequationum of 1615 (§ 178), in Bachet's edition of Diophantus
of 1621, in Camillo Glorioso in 1627 (§ 196). In numerical equations
Oughtred uses / for x, but the small letters q, c, qq, gc, etc., for the
higher powers of x (§ 181). Sometimes Oughtred employs also the
corresponding capital letters. Descartes very often used, in his corre-
spondence, notations different from his own, as perhaps more familiar
to his correspondents than his own. Thus, as late as 1640, in a letter
to Mersenne (September 30, 1640), Descartes1 writes "1C-6N = 40,"
which means xz — 6# = 40. In the Regulae ad directionem ingenii,
Descartes represents2 by a, b, c, etc., known magnitudes and by
A, J5, C, etc., the unknowns; this is the exact opposite of the use of
these letters found later in Rahn.
Crossed numerals representing powers of unknowns. — Interest-
ing is the attitude of P. A. Cataldi of Bologna, who deplored the
existence of many different notations in different countries for the un-
known numbers and their powers, and the inconveniences resulting
from such diversity. He points out also the difficulty of finding in the
ordinary printing establishment the proper type for the representation
of the different powers. He proposes3 to remove both inconveniences
by the use of numerals indicating the powers of the unknown and dis-
tinguishing them from ordinary numbers by crossing them out, so
that 0, *, 2, 3, • • • . , would stand for x°, x', x2, x3 Such crossed
numerals, he argued, were convenient and would be found in printing
offices since they are used in arithmetics giving the scratch method of
dividing, called by the Italians the "a Galea" method. The reader
will recall that Cataldi's notation closely resembles that of Leonard
1 CEuvres de Descartes, Vol. Ill (1899), p. 190, 196, 197; also Vol. XII, p. 279.
2 Op. cit.y Vol. X (1908), p. 455, 462.
3 P. A. Cataldi, Trattato dell' algebra proportionate (Bologna, 1610), and in his
later works. See G. Wertheim in Bibliotheca mathemalica (3d ser.), Vol. II (1901),
p. 146, 147.
UNKNOWN NUMBERS 381
and Thomas Digges in England (§ 170). These symbols failed of
adoption by other mathematicians. We have seen that in 1627
Camillo Glorioso, in a work published at Naples/ wrote N for x,
and q, c, qq, qc, cc, qqc, qcc, and ccc for x2, x3, . . . . , z9, respectively
(§ 196). In 1613 Glorioso had followed Stevin in representing an un-
known quantity by 1O.
340. Descartes' z, y, x. — The use of z, y, x . . . . to represent un-
knowns is due to Rcn6 Descartes, in his La geometric (1637). Without
comment, he introduces the use of the first letters of the alphabet to
signify known quantities and the use of the last letters to signify
unknown quantities. His own language is: "... 1'autre, LN, est \a la
moitie de 1'autre quantity connue, qui estoit multipliee par z, quo ie
suppose estre la ligne inconnue."2 Again: "... ie considere ... Quo
le segment de la ligne AB, qui est entre les poins A et B, soit nomine x,
et que BC soit nomme' y; ... la proportion qui est entre les cost^s AB
et BR est aussy donntfc, et ie la pose cornme de z a 6; de fagon qu' A B
estant x, RB sera — , et la toute CR sera y-\ — . ..." Later he says:
z z
"et pour ce que CB et BA sont deux quantity's indetcrmme*es et in-
connues, ie les nomme, Tune y; et 1'autre x. Mais, affin de trouver le
rapport de Tune a 1'autre, ie considere aussy les quantites connues qui
deterrninent la description de cete ligne courbe: comrne GA que jo
nomine a, KL que je nomme b, et NL, parallele a GA, que ie nomine
GV'3 As co-ordinates he uses later only x and y. In equations, in the
third book of the Geometric, x predominates. In manuscripts written
in the interval 1629-40, the unknown z occurs only once.4 In the other
places x and y occur. In a paper on Cartesian ovals,5 prepared before
1629, x alone occurs as unknown, y being used as a parameter. This
is the earliest place in which Descartes used one of the last letters of
the alphabet to represent an unknown. A little later he used x, y, z
again as known quantities.6
Some historical writers have focused their attention upon the x,
disregarding the y and z, and the other changes in notation made by
1 Camillo Gloriosi, Exercitationes mathematical, decas I (Naples, 1627). Also
Ad theorema geometricvm, d nobilissimo viro proposition, Joannis Camilli Gloriosi
(Venice, 1613), p. 26. It is of interest that Glorioso succeeded Galileo in the mathe-
matical chair at Padua.
2 (Euvres de Descartes, Vol. VI (1902), p. 375.
3 Ibid., p. 394.
4 Ibid., Vol. X, p. 288-324.
5 Ibid., p. 310. 6 Ibid., p. 299.
382 A HISTORY OF MATHEMATICAL NOTATIONS
Descartes; these writers have endeavored to connect this x with older
symbols or with Arabic words. Thus, J. Tropfke,1 P. Treutlein,2 and
M. Curtze8 advanced the view that the symbol for the unknown used
by early German writers, 2£, looked so much like an x that it could
easily have been taken as such, and that Descartes actually did inter-
pret and use it as an x. But Descartes' mode of introducing the
knowns a, b, c, etc., and the unknowns z, T/, x makes this hypothesis
improbable. Moreover, G. Enestrom has shown4 that in a letter of
March 26, 1619, addressed to Isaac Beeckman, Descartes used the
symbol 3£ as a symbol in form distinct from x, hence later could not
have mistaken it for an 3£» At one time, before 1637, Descartes5 used
x along the side of 3£; at that time x, y, z are still used by him as
symbols for known quantities. German symbols, including the 2£ for
x, as they are found in the algebra of Clavius, occur regularly in a
manuscript6 due to Descartes, the Opuscules de 1619-1621.
All these facts caused Tropfke in 1921 to abandon his old view7 on
the origin of x, but he now argues with force that the resemblance of
x and 2£, and Descartes' familiarity with 2£, may account for the
fact that in the latter part of Descartes' Geometric the x occurs more
frequently than z and y. Enestrom, on the other hand, inclines to
the view that the predominance of x over y and z is due to typo-
graphical reasons, type for x being more plentiful because of the more
frequent occurrence of the letter x, to y and 2, in the French and Latin
languages.8
There is nothing to support the hypothesis on the origin of x
due to Wertheim,9 namely, that the- Cartesian x is simply the nota-
tion of the Italian Cataldi who represented the first power of the
unknown by a crossed "one," thus Z. Nor is there historical evidence
1 J. Tropfke, Geschichte der Elementar-Mathematik, Vol. I (Leipzig, 1902), p.
150.
2 P. Treutlein, "Die deutsche Coss," Abhandl. z. Geschichte d. mathematischen
Wins., Vol. II (1879), p. 32.
3 M. Curtze, ibid., Vol. XIII (1902), p. 473.
4 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. VI (1905), p. 316, 317,
405, 406. See also his remarks in ibid. (1884) (Sp. 43); ibid. (1889), p. 91. ,Jhe
letter to Beeckman is reproduced in (Euvres de Descartes, Vol. X (1908), p. 155.
6 (Euvres de Descartes, Vol. X (Paris, 1908), p. 299. See also Vol. Ill, Appendix
II, No. 480.
6 Ibid., Vol. X (1908), p. 234.
7 J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 44-46.
8 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. VI, p. 317.
9 G. Enestrom, ibid.
UNKNOWN NUMBERS 383
to support the statement found in Noah Webster's Dictionary, under
the letter x, to the effect that "x was used as an abbreviation of Ar.
shei a thing, something, which, in the Middle Ages, was used to desig-
nate the unknown, and was then prevailingly transcribed as xei."
341. Spread of Descartes' signs. — Descartes' x, y, and z notation
did not meet with immediate adoption. J. II. Rahn, for example,
says in his Teutsche Algebra (1659): "Descartes' way is to signify
known quantities by the former letters of the alphabet, and unknown
by the latter [z, y, x> etc.]. But I choose to signify the unknown quan-
tities by small letters and the known by capitals." Accordingly, in a
number of his geometrical problems, Rahn uses a and A, etc., but in
the book as a whole he uses z, y, x freely.
As late as 1670 the learned bishop, Johann Caramuel, in his Mathe-
sis biceps ... , Campagna (near Naples), page 123, gives an old nota-
tion. He states an old problem and gives the solution of it as found in
Geysius; it illustrates the rhetorical exposition found in some books as
late as the time of Wallis, Newton, and Leibniz. We quote: "Dicebat
Augias Herculi: Meorum armentorurn media pars est in tali loco
octavi in tali, decirna in tali, 20™a in tali 60™a in tali, & 50 . sunc hie.
Et Geysius libr. 3 Cossa Cap. 4. haec pecora numeraturus sic scribit.
"Finge 1. a. partes |a, |a, -^a, ^Va> «Va & additae (hoc est, in
summam reductae) sunt |fa & quibus de 1. a. sublatis, restant <& a
aequalia 50. Jam, quia sictus, est fractio, multiplicando reducatur,
& 1. a. aequantur 240. Hie est numerus pecorum Augiae." ("Augias
said to Hercules: 'Half of my cattle is in such a place, | in such, tV m
such, -sV m such, uV in such, and here there are 50. And Geysius in
Book 3, Cossa, Chap. 4, finds the number of the herd thus: Assume
La., the parts are Ja, ^a, -^a, ^a, -^a, and these added [i.e., reduced
to a sum] are ||a which subtracted from 1. a, leaves ^4a, equal to 50.
Now, the fraction is removed by multiplication, and 1. a equal 240.
This is the number of Augias' herd/ ")
Descartes' notation, x, y, z, is adopted by Gerard Kinckhuysen,1
in his Algebra (1661). The earliest systematic use of three co-ordinates
in analytical geometry is found in De la Hire, who in his Nouveaux
Siemens des sections coniques (Paris, 1679) employed (p. 27) x, y, v.
A. Parent2 used x, y, z; Euler3, in 1728, ty x, y; Joh. Bernoulli,4 in 1715,
1 Gerard Kinckhuysen, Algebra ofte Slel-Konst (Haerlcm, 1661), p. 6.
2 A. Parent, Essais et recherches de math, et de phys., Vol. I (Paris, 1705).
3 Euler in Comm. Aca. Petr., II, 2, p. 48 (year 1728, printed 1732).
4 Leibniz and Bernoulli, Commerdum philosophicum et mathematicum, Vol. II
(1745), p. 345.
384 A HISTORY OF MATHEMATICAL NOTATIONS
x, yy z in a letter (February 6, 1715) to Leibniz. H. Pitot1 applied the
three co-ordinates to the helix in 1724.
SIGNS OF AGGREGATION
342. Introduction. — In a rhetorical or syncopated algebra, the
aggregation of terms could be indicated in words. Hence the need for
symbols of aggregation was not urgent. Not until the fifteenth and
sixteenth centuries did the convenience and need for such signs
definitely present itself. Various devices were invoked: (1) the hori-
zontal bar, placed below or above the expression affected; (2) the
use of abbreviations of words signifying aggregation, as for instance u
or v for universalis or vniversalis, which, however, did not always indi-
cate clearly the exact range of terms affected ; (3) the use of dots or
commas placed before the expression affected, or at the close of such
an expression, or (still more commonly) placed both before and after;
(4) the use of parentheses (round parentheses or brackets or braces).
Of these devices the parentheses were the slowest to find wide adop-
tion in all countries, but now they have fairly won their place in
competition with the horizontal bar or vinculum. Parentheses pre-
vailed for typographical reasons. Other things being equal, there is a
preference for symbols which proceed in orderly fashion as do the
letters in ordinary printing, without the placing of signs in high or low
positions that would break a line into two or more sublines. A vincu-
lum at once necessitates two terraces of type, the setting of which
calls for more time and greater technical skill. At the present time
1 H. Pitot, Memoires de V academic d. scien., ann6e 1724 (Paris, 1726). Taken
from H. W ieleitner, Geschichte der Mathematik, 2. Teil, 2. Halfte (Berlin und Leipzig,
1921), p. 92.
To what extent the letter x has been incorporated in mathematical language
is illustrated by the French expression Strefort en x, which means "being strong in
mathematics." In the same way, fate d x means "a mathematical head." The
French give an amusing "demonstration" that old men who were tbte d x never
were pressed into military service so as to have been conscripts. For, if they were
conscripts, they would now be ex-conscripts. Expressed in symbols we would have
Ox = ear-conscript.
Dividing both sides by x gives
0=e-conscript.
Dividing now by e yields
conscript =-.
According to this, the conscript would be la tbte assurSe (i.e., 0 over e, or, the head
assured against casualty), which is absurd.
AGGREGATION 385
the introducing of typesetting machines and the great cost of type-
setting by hand operate against a double or multiple line notation.
The dots have not generally prevailed in the marking of aggregation
for the reason, no doubt, that there was danger of confusion since dots
are used in many other symbolisms — those for multiplication, division,
ratio, decimal fractions, time-derivatives, marking a number into
periods of two or three digits, etc.
343. Aggregation expressed by letters. — The expression of aggrega-
tion by the use of letters serving as abbreviations of words expressing
aggregation is not quite as old as the use of horizontal bars, but it is
more common in works of the sixteenth century. The need of marking
the aggregation of terms arose most frequently in the treatment of
radicals. Thus Pacioli, in his Summa of 1494 and 1523, employs v
(vniversale) in marking the root of a binomial or polynomial (§ 135).
This and two additional abbreviations occur in Cardan (§ 141). The
German manuscript of Andreas Alexander (1524) contains the letters
cs for communis (§325); Chr. Rudolff sometimes used the word
"collect," as in "i/ des collects 17+1/208" to designate >/17+T/208.1
J. Scheubel adopted Ha. col. (§ 159). S. Stevin, Fr. Vieta, and A.
Romanus wrote bin., or bino., or binomia, trinom., or similar abbrevia-
tions (§ 320). The u or v is found again in Pedro Nunez (who uses also
L for "ligature"),2 Leonard and Thomas Digges (§ 334), in J. R.
Brasser3 who in 1663 lets v signify "universal radix" and writes
"#l/.8 -T- i/45" to represent ^8 — 1/45. W. Oughtred sometimes wrote
1/u or i/b (§§ 183, 334). In 1685 John Wallis4 explains the notations
1/6:2+1/3, i/r:2-i/3, i/u:2±i/3, 1/2 ±1/3, i/:2±l/3, where b
means "binomial," u "universal," r "residual," and sometimes uses
redundant forms like \/b : i/5+ 1 : .
344. Aggregation expressed by horizontal bars or vinculums. — The
use ef the horizontal bar to express the aggregation of terms goes back
to the time of Nicolas Chuquet who in his manuscript (1484) under-
lines the parts affected (§ 130). We have seen that the same idea is
followed by the German Andreas Alexander (§ 325) in a manuscript of
1545, and by the Italian Raffaele Bombelli in the manuscript edition
1 J. Tropfke, op. tit., Vol. II (1921), p. 150.
2 Pedro Nufiez, Libra de algebra en arithmetica y geometria (Anvers, 1507),
fol. 52.
1 J. R. Brasser, Regula of Algebra (Amsterdam, 1663), p. 27.
* John Wallis, Treatise of Algebra (London, 1685), p. 109, 110. The uso of
letters for aggregation practically disappeared in the seventeenth century.
386 A HISTORY OF MATHEMATICAL NOTATIONS
of his algebra (about 1550) where he wrote ^2 + V — 121 in this
manner i1 R*[2 . p . R[Q m 121]] ; parentheses were used and, in addition,
vinculums were drawn underneath to indicate the range of the paren-
theses. The employment of a long horizontal brace in connection with
the radical sign was introduced by Thomas Harriot2 in 1631; he
expresses aggregation thus: \/ccc+i/cccccc-~bbbbbb. This notation
may, perhaps, have suggested to Descartes his new radical sym-
bolism of 1637. Before that date, Descartes had used dots in the man-
ner of Stif el and Van Ceulen. He wrote8 1/ . 2 - >/2 . f or 1/2 - 1/2. He
attaches the vinculum to the radical sign j/ and writes V/a2+bz)
V — -|a+l/Jaa+66, and in case of cube roots *C. %q+Vlgg— 27p3.
Descartes does not use parentheses in his Geometric. Descartes uses
the horizontal bar only in connection with the radical sign. Its
general use for aggregation is due to Fr, van Schooten, who, in edit-
ing Vieta's collected works in 1646, discarded parentheses and placed
a horizontal bar above the parts affected. Thus Van Schooten's "B in
D quad.+B in D" means B(D*+BD). Vieta4 himself in 1593 had
written this expression differently, namely, in this manner:
" . (D. quadratwn "
B in < , o . ~
\+B in D
B. Cavalieri in his Geometria indivisibilibae and in his Exercitationes
geometriae sex (1647) uses the vinculum in this manner, AB, to indi-
cate that the two letters A and B are not to be taken separately, but
conjointly, so as to represent a straight line, drawn from the point A
to the point B.
Descartes' and Van Schooten's stressing the use of the vinculum
led to its adoption by J. Prestet in his popular text, Siemens des
Mathtmatiques (Paris, 1675). In an account of Rolle5 the cube root
is to be taken of 2+#.-121, i.e., of 2+1/-121. G. W. Leibniz6 in a
1 See E. Bertolotti in Scientia, Vol. XXXIII (1923), p. 391 n.
2 Thomas Harriot, Arlis analyticae praxis (London, 1631), p. 100.
3 R. Descartes, (Euvres (6d. Ch. Adam et P. Tannery), Vol. X (Paris, 1908),
p. 286 f., also p. 247, 248.
4 See J. Tropfke, op. tit., Vol. II (1921), p. 30.
5 Journal des Sgavans de Tan 1683 (Amsterdam, 1709), p. 97.
8 G. W. Leibniz* letter to D. Oldenburgh, Feb. 3, 1672-73, printed in J.
Collin's Commercium epistolicum (1712).
AGGREGATION 387
letter of 1672 uses expressions like aco6co6oocc/>&<x>c</>cood, where co
signifies "difference." Occasionally he uses the vinculum until about
1708, though usually he prefers round parentheses. In 1708 Leibniz'
preference for round parentheses (§ 197) is indicated by a writer in the
Ada eruditorum. Joh. (1) Bernoulli, in his Lectiones de calculo differ-
entialium, uses vinculums but no parentheses.1
345. In England the notations of W. Oughtred, Thomas Harriot,
John Wallis, and Isaac Barrow tended to retard the immediate intro-
duction of the vinculum. But it was used freely by John Kersey
(1673)2 who wrote y (2) : |r — \/\rr— s : and by Newton, as, for in-
stance, in his letter to D. Oldenburgh of June 13, 1676, where he gives
the binomial formula as the expansion of P+PQ\ n . In his De
Analysi per Aequationes numero terminorum Infinitas, Newton writes3
— 12x?/-fl7 = 0 to represent {[(y—4)y — 5]y—12}y+17
= 0. This notation was adopted by Edmund Halley,4 David Gregory,
and John Craig; it had a firm foothold in England at the close of the
seventeenth century. During the eighteenth century it was the regular
symbol of aggregation in England and France; it took the place very
largely of the parentheses which are in vogue in our day. The vincu-
lum appears to the exclusion of parentheses in the Geometria organica
(1720) of Colin Maclaurin, in the Elements of Algebra of Nicholas
Saunderson (Vol. I, 1741), in the Treatise of Algebra (2d ed.; London,
1756) of Maclaurin. Likewise, in Thomas Simpson's Mathematical
Dissertations (1743) and in the 1769 London edition of Isaac Newton's
Universal Arithmetick (translated by Ralphson and revised by Cunn),
vinculums are used and parentheses do not occur. Some use of the
vinculum was made nearly everywhere during the eighteenth century,
especially in connection with the radical sign -j/, so as to produce \/ .
This last form has maintained its place down to the present time.
However, there are eighteenth-century writers who avoid the vincu-
lum altogether even in connection with the radical sign, and use
1 The Johannis (1) Bernoulli! Lectiones de calculo differentialium, which re-
mained in manuscript until 1922, when it was published by Paul Schafheitlin
in Verhandlungen der Naturforschenden Gesellschaft in Basel, Vol. XXXIV
(1922).
2 John Kersey, Algebra (London, 1673), p. 55.
3 Commercium epistolicum (6d. Biot et Lcfort; Paris, 1856), p. 63.
4 Philosophical Transactions (London), Vol. XV-XVI (1684-91), p. 393; Vol.
XIX (1695-97), p. 60, 645, 709.
388 A HISTORY OF MATHEMATICAL NOTATIONS
parentheses exclusively. Among these are Poleni (1729),1 Cramer
(1750) ,2 and Cossali (1797) .3
346. There was considerable vacillation on the use of the vinculum
in designating the square root of minus unity. Some authors wrote
\/ — 1; others wrote I/— 1 or j/(— !)• For example, >/— 1 was the
designation adopted by J. Wallis,4 J. d'Alembert,6 1. A. Segner,6 C. A.
Vandermonde,7 A. Fontaine.8 Odd in appearance is an expression of
Euler,9 v/(2v/-l-4). But i/(-l) was preferred by Du S<5jour10 in
1768 and by Waring11 in 1782; j/-l by Laplace12 in 1810.
347. It is not surprising that, in times when a notation was passing
out and another one taking its place, cases should arise where both are
used, causing redundancy. For example, J. Stampioen in Holland
sometimes expresses aggregation of a set of terms by three notations,
any one of which would have been sufficient; he writes13 in 1640,
I/. (aaa-\-6aab+9bba), where the dot, the parentheses, and the vincu-
lum appear; John Craig14 writes V/2ay — y2: and v : V/6a4— fa2, where
the colon is the old Oughtredian sign of aggregation, which is here
superfluous, because of the vinculum. Tautology in notation is found
in Edward Cocker15 in expressions like ^aa+bb, V :c+\bb — \b, and
1 loannis Poleni, Epistolarvm mathematicarvm fascicvlvs (Padua, 1729).
2 Gabriel Cramer, L' Analyse des lignes courbes algebriques (Geneva, 1750).
3 Pietro Cossali, Origini ... dell'algebra, Vol. I (Parma, 1797).
4 John Wallis, Treatise of Algebra (London, 1685), p. 266.
6 J. d'Alembert in Histoire de Facadtmie r. des sciences, anne*e 1745 (Paris,
1749), p. 383.
6 1. A. Segner, Cursus mathemalici, Pars IV (Halle, 1763), p. 44.
7 C. A. Vandermonde in op. cit., anne*e 1771 (Paris, 1774), p. 385.
8 A. Fontaine, ibid., annee 1747 (Paris, 1752), p. 607.
9 L. Euler in Histoire de Vacademie r. d. sciences et des belles lettres, ann6e 1749
(Berlin, 1751), p. 228.
10 Du S6jour, ibid. (1768; Paris, 1770), p. 207.
11 E. Waring, Meditationes algebraicae (Cambridge; 3d ed., 1782), p. xxxvl,
etc.
13 P. S. Laplace in Memoires d. Vacademie r. d. sciences, anne*e 1817 (Paris, 1819),
p. 153.
11 /. /. Stampionii Wis-Konsligh ende Reden-Maetigh Bewijs (The Hague,
1640), p. 7.
14 John Craig, Philosophical Transactions, Vol. XIX (London, 1695-97), p. 709.
15 Cockers Artificial Arithmelick Composed by Edward Cocker Pe-
rused, corrected and published by John Hawkins (London, 1702) ['To the Reader,"
1684], p. 368, 375.
AGGREGATION 389
a few times in John Wallis.1 In the Ada eruditorum (1709), page 327,
one finds n7/V/a = ||/[(x— nna)3], where the [ ] makes, we believe, its
first appearance in this journal, but does so as a redundant symbol.
348. Aggregation expressed by dots. — The denoting of aggregation
by placing a dot before the expression affected is first encountered in
Christoff Rudolff (§ 148). It is found next in the Arithmetica Integra
of M. Stifel, who sometimes places a dot also at the end. He writes2
1/2.12+i/z 6+.1/2- 12-1/2 6 for our V12+V/6+^12-l6; also
1/0.144-6+1/^.144-6 for 1/114^+1/144^6. In 1605 C. Di-
buadius3 writes v'.2-i/.2+j/.2+v'.2+j/.2+v/2 as the side of
a regular polygon of 128 sides inscribed in a circle of unit radius, i.e.,
\2-\2+V2+'V2+v/2+T/2 (see also §332). It must be ad-
mitted that this old notation is simpler than the modern. In SnelPs
translation4 into Latin (1610) of Ludolph van Ceulen's work on the
circle is given the same notation, i/.2+i/.2— 1/.2 — i/.2+i/2| —
1/2J. In SnelFs 1615 translation6 into Latin of Ludolph's arithmetic
and geometry is given the number i/.2 — 1/.2J + 1/1J which, when
divided by j/.2+j/.2£+f/lJ, gives the quotient i/5+l-j/.5+
1/20. The Swiss Joh. Arduser6 in 1646 writes i/.2-f-i/-2+i/.2+
I/. 2+ 1/2+ 1/. 2+ 1/. 2+ 1/3, etc., as the side of an inscribed poly-
gon of 768 sides, where -f- means "minus."
The substitution of two dots (the colon) in the place of the single
dot was effected by Oughtred in the 1631 and later editions of his
Clavis mathematicae. With him this change became necessary when
he adopted the single dot as the sign of ratio. He wrote ordinarily
l/q:BCq—BAq: for ^BC2— BA2, placing colons before and after
the terms to be aggregated (§ 181). 7
1 John Wallis, Treatise of Algebra (London, 1685), p. 133.
2 M. Stifel, Arithmetica integra (Nurnbcrg, 1544), fol. 135t>°. See J. Tropfke,
op. cit., Vol. Ill (Leipzig, 1922), p. 131.
3 C. Dibvadii in arithmeticam irrationalivm Evclidis decimo elernentorum libro
(Arnhem, 1605).
4 Willebrordus Snellius, Lvdolphi a Cevlen de Circvlo et adscriptis liber ...6
vernaculo Latina fecit ... (Leyden, 1610), p. 1, 5.
6 Fvndamenta arithmetica et geometrica. ... Lvdolpho a Cevlen , ... in Lalinum
translata a Wil. Sn. (Leyden, 1615), p. 27,
6 Joh. Arduser, Geomelriae theoricae et practicae XII libri (Zurich, 1646),
fol. 1816.
7 W Oughtred, Clavis mathematicae (1652), p. 104.
390 A HISTORY OF MATHEMATICAL NOTATIONS
Sometimes, when all the terms to the end of an expression are to
be aggregated, the closing colon is omitted. In rare instances the
opening colon is missing. A few times in the 1694 English edition, dots
take the place of the colon. Oughtred's colons were widely used in
England. As late as 1670 and 1693 John Wallis1 writes i/:5-2v/3:.
It occurs in Edward Cocker's2 arithmetic of 1684, Jonas Moore's
arithmetic3 of 1688, where C:A+E means the cube of (A-\-E). James
Bernoulli4 gives in 1689 |/:a+i/:a+i/:a+i/:a-fj/:a-K etc.
These methods of denoting aggregation practically disappeared at the
beginning of the eighteenth century, but in more recent time they
have been reintroduced. Thus, R. Carmichacl5 writes in his Calculus of
Operations: "D. uv~u. Dv-\-Du. v." G, Peano has made the proposal
to employ points as well as parentheses.6 He lets a. be be identical
with a(6c), aibc.d with a[(bc)d]y ab.cdie.fg .'. hk.l with {[(ab)(cd)]
349. Aggregation expressed by commas. — An attempt on the part of
Hdrigone (§ 189) and Leibniz to give the comma the force of a symbol
of aggregation, somewhat similar to Rudolffs, Stifel's, and van
Ceulen's previous use of the dot and Oughtred's use of the colon, was
not successful. In 1702 Leibniz7 writes c — b} I for (c — b)l, and c— 6,
d— 6, I for (c — b)(d—b)L In 1709 a reviewer8 in the Ada eruditorum
represents (m-±[m—l])x(m~'l)*m by (m,:m— l)xw~1':m, a designation
somewhat simpler than our modern form.
350. Aggregation expressed by parenthesis is found in rare in-
stances as early as the sixteenth century. Parentheses present com-
paratively no special difficulties to the typesetter. Nevertheless, it
took over two centuries before they met with general adoption as
mathematical symbols. Perhaps the fact that they were used quite
extensively as purely rhetorical symbols in ordinary writing helped to
1 John Wallis in Philosophical Transactions, Vol. V (London, for the year
1670), p. 2203; Treatise of Algebra (London, 1685), p. 109; Latin ed. (1693), p. 120.
2 Cocker's Artificial Arithmetick .... perused .... by John Hawkins (Lon-
don, 1684), p. 405.
3 Moore's Arithmetick: in Four Books (London, 1688; 3d ed.), Book IV, p. 425.
4 Positiones arithmeticae de seriebvs infinilis .... Jacobo Bernoulli (Basel,
1689).
6 R. Carmichael, Der Operationscalcul, deutsch von C. H. Schnuse (Braun-
schweig, 1857), p. 16.
6 G. Peano, Formulaire mathemalique, fidition de Fan 1902-3 (Turin, 1903),
p. 4.
7 G. W. Leibniz in Ada eruditorum (1702), p. 212.
8 Reviewer in ibid. (1709), p. 230. See also p. 180.
AGGREGATION 391
retard their general adoption as mathematical symbols. John Wallis,
for example, used parentheses very extensively as symbols containing
parenthetical rhetorical statements, but made practically no use of
them as symbols in algebra.
As a rhetorical sign to inclose an auxiliary or parenthetical state-
ment parentheses are found in Newton's De analysi per equaliones
numero lerminorum infinilas, as given by John Collins in the Com-
merdum epislolicum (1712). In 1740 De Gua1 wrote equations in the
running text and inclosed them in parentheses; he wrote, for example,
"... seroit (7 a — 3x • dx = 3l/2az — xx • dx) et oil Tare de cercle. ..."
English mathematicians adhered to the use of vinculums, and of
colons placed before and after a polynomial, more tenaciously than
did the French; while even the French were more disposed to stress
their use than were Leibniz and Euler. It was Leibniz, the younger
Bernoullis, and Euler who formed the habit of employing parentheses
more freely and to resort to the vinculum less freely than did other
mathematicians of their day. The straight line, as a sign of aggrega-
tion, is older than the parenthesis. We have seen that Chuquet, in his
Triparly of 1484, underlined the terms that were to be taken together.
351. Early occurrence of parentheses. — Brackets2 are found in the
manuscript edition of R. Bombelli's Algebra (about 1550) in the
expressions like &[2mR[Om.l2l]] which stands for ^2-l/-121.
In the printed edition of 1572 an inverted capital letter L was employed
to express radix legata; see the facsimile reproduction (Fig. 50).
Michael Stifel does not use parentheses as signs of aggregation in his
printed works, but in one of his handwritten marginal notes3 occurs
the following: ". . . . faciant aggregatum (12 — j/44) quod sumptum
cum (v/44 — 2) facial 10" (i.e., "....One obtains the aggregate
(12 — -J/44), which added to (j/44 — 2) makes 10"). It is our opinion
that these parentheses are punctuation marks, rather than mathe-
matical symbols; signs of aggregation are not needed here. In the
1593 edition of F. Vieta's Zetelica, published in Turin, occur braces
and brackets (§ 177) sometimes as open parentheses, at other times
as closed ones. In Vieta's collected works, edited by Fr. van Schooten
1 Jean Paul de Gua de Malves, Usages de V analyse de Descartes (Paris, 1740),
p. 302.
2 See E. Bortolotti in Scientia, Vol. XXXIII (1923), p. 390.
3E. Hoppe, "Michael Stifels handschriftlicher Nachlass," Mitteilungen
Math. Gesellschaft Hamburg, III (1900), p. 420. See J. Tropfke, op. cii.9 Vol. II
(2d ed., 1921), p. 28, n. 114.
392 A HISTORY OF MATHEMATICAL NOTATIONS
in 1646, practically all parentheses are displaced by vinculums. How-
ever, in J. L. de Vaulezard's translation1 into French of Vieta's
Zetetica round parentheses are employed. Round parentheses are en-
countered in Tartaglia,2 Cardan (but only once in his Ars Magnci)?
Clavius (see Fig. 66), Errard de Bar-le-Duc,4 Follinus,5 Girard,6
Norwood,7 Hume,8 Stampioen, Henrion, Jacobo de Billy,9 Renaldini10
and Foster.11 This is a fairly representative group of writers using
parentheses, in a limited degree; there are in this group Italians, Ger-
mans, Dutch, French, English. And yet the mathematicians of none
of the countries represented in this group adopted the general use of
parentheses at that time. One reason for this failure lies in the fact
that the vinculum, and some of the other devices for expressing ag-
gregation, served their purpose very well. In those days when machine
processes in printing were not in vogue, and when typesetting was
done by hand, it was less essential than it is now that symbols should,
in orderly fashion, follow each other in a line. If one or more vincu-
lums were to be placed above a given polynomial, such a demand
upon the printer was less serious in those days than it is at the present
time.
1 J. L. de Vaulezard's Zetettques de F. Viete (Paris, 1630), p. 218. Reference
taken from the Encyclopedic d. scien. math., Tom I, Vol. I, p. 28.
2 N. Tartaglia, General trattato di numeri e misure (Venice), Vol. II (1556), fol.
1676, 1696, 1706, 1746, 177a, etc., in expressions like "&v.(ft 28 men R 10)" for
VV28-I/10; f°l- 168&» "men (22 men #6" for -(22 -1/6), only the opening
part being used. See G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. VII
(1906-7), p. 296. Similarly, in La Quarta Parte del general trattato (1560), fol.
40#, he regularly omits the second part of the parenthesis when occurring on the
margin, but in the running text both parts occur usually.
3 H. Cardano, Ars magna, as printed in Opera, Vol. IV (1663), fol. 438.
4 1. Errard de Bar-le-Duc, La geometric et practique generate d'icelle (3d ed.;
revue* par D. H. P. E. M.; Paris, 1619), p. 216.
6 Hermann Follinus, Algebra sive liber de rebus occvltis (Cologne; 1622), p. 157.
8 A. Girard, Invention nouvclle en Valgebre (Amsterdam, 1629), p. 17.
7 R. Norwood, Trigonometric (London, 1631), Book I, p. 30.
8 Jac. Humius, Traite de Valgebre (Paris, 1635).
9 Jacobo de Billy, Novae geometriae clavis algebra (Paris, 1643), p. 157; also
in an Abridgement of the Precepts of Algebra (written in French by James de
Billy; London, 1659), p. 346.
10 Carlo Renaldini, Opus algebricum (1644; enlarged edition, 1665). Taken from
Ch. Hutton, Tracts on Mathematical and Philosophical Subjects, Vol. II (1812),
p. 297.
11 Samuel Foster, Miscellanies: or Mathematical Lucubrations (London, 1659),
p. 7.
AGGREGATION 393
And so it happened that in the second half of the seventeenth
century, parentheses occur in algebra less frequently than during the
first half of that century. However, voices in their favor are heard.
The Dutch writer, J. J. Blassiere,1 explained in 1770 the three nota-
tions (2a+56)(3a-4&), (2a+56)X(3a-46), and 2a+MX3a^4b,
and remarked: "Mais comme la premiere mani£re de les enfermer
entre des Parentheses, est la moins sujette a erreur, nous nous en
servirons dans la suite." E. Waring in 17622 uses the vinculum but no
parentheses; in 17823 he employs parentheses and vinculurns inter-
changeably. Before the eighteenth century parentheses hardly ever
occur in the Philosophical Transactions of London, in the publications
of the Paris Academy of Sciences, in the A eta eruditorum published in
Leipzig. But with the beginning of the eighteenth century, paren-
theses do appear. In the Ada eruditorum, Carre4 of Paris uses them
in 1701, G. W. Leibniz5 in 1702, a reviewer of Gabriele Manfredi6 in
1708. Then comes in 1708 (§ 197) the statement of policy7 in the Acta
eruditorum in favor of the Leibnizian symbols, so that "in place of
l/aa+bb we write i/(aa+bb) and for aa+bbXc we write aa+bb,c
.... we shall designate aa+bbm by (aa-\-bb)m: whence Vaa+bb will
m/
be =(aa+bb)1:m and v aa+bbn=(aa+bb)n:m. Indeed, we do not
doubt that all mathematicians reading these Acta recognize the pre-
eminence of Mr. Leibniz' symbolism and agree with us in regard to it."
From now on round parentheses appear frequently in the Acta
eruditorum. In 1709 square brackets make their appearance.8 In the
Philosophical Transactions of London9 one of the first appearances of
parentheses was in an article by the Frenchman P. L. Maupertuis
in 1731, while in the Histoire de Vacademie royale des sciences in Paris,10
1 J. J. Blassiere, Institution du calcul numerique et litteral. (a la Haye, 1770),
2. Partie, p. 27.
2 E. Waring, Miscellanea analytica (Cambridge, 1762).
3E. Waring, Meditationes algebraicae (Cambridge; 3d ed., 1782).
<L. Carre* in Acta eruditorum (1701), p. 281.
« G. W. Leibniz, ibid. (1702), p. 219.
6 Gabriel Manfredi, ibid. (1708), p. 268.
7 Ibid. (1708), p. 271.
s Ibid. (1709), p. 327.
9 P. L. Maupertuis in Philosophical Transactions, for 1731-32, Vol. XXXVII
(London), p. 245.
10 Johann II Bernoulli, Histoire de Vacadtmie royale des sciences, ann6e 1732
(Paris, 1735), p. 240 ff.
394 A HISTORY OF MATHEMATICAL NOTATIONS
Johann (John) Bernoulli of Bale first used parentheses and brackets in
the volume for the year 1732. In the volumes of the Petrograd
Academy, J. Hermann1 uses parentheses, in the first volume, for the
year 1726; in the third volume, for the year 1728, L. Euler2 and
Daniel Bernoulli used round parentheses and brackets.
352. The constant use of parentheses in the stream of articles from
the pen of Euler that appeared during the eighteenth century con-
tributed vastly toward accustoming mathematicians to their use.
Some of his articles present an odd appearance from the fact that the
closing part of a round parenthesis is much larger than the opening
?t Z \
part,1 as in (1 — ) (1 ). Daniel Bernoulli4 in 1753 uses round
7T 7T S j
parentheses and brackets in the same expression while T. U. T,
Aepinus5 and later Euler use two types of round parentheses of this
sort, C(P+y)(M-l)+AM3. In the publications of the Paris
Academy, parentheses are used by Johann Bernoulli (both round and
square ones),6 A. C. Clairaut,7 P. L. Maupertuis,8 F. Nicole,9 Ch. de
Montigny,10 Le Marquis de Courtivron,11 J. d'Alembert,12 N. C. de
Condorcet,13 J. Lagrange.14 These illustrations show that about the
middle of the eighteenth century parentheses were making vigorous
inroads upon the territory previously occupied in France by vincu-
lums almost exclusively.
1 J. Hermann, Commentarii academiae sdentiarum imperialis Petropolitanae,
Tomus I ad annum 1726 (Petropoli, 1728), p. 15.
2 Ibid., Tomus III (1728; Petropoli, 1732), p. 114, 221.
3 L. Euler in Miscellanea Berolinensia, Vol. VII (Berlin, 1743), p. 93, 95, 97,
139, 177.
4 D. Bernoulli in Histoire de I'academie r. des sciences et belles lettres, anne*e 1753
(Berlin, 1755), p. 175.
5 Aepinus in ibid., anne"e 1751 (Berlin, 1753), p. 375; anne*e 1757 (Berlin,
1759), p. 308-21.
6 Histoire de Vacademie r. des sciences, anne*e 1732 (Paris, 1735), p. 240, 257.
7 Ibid., ann&j 1732, p. 385, 387.
8 Ibid., ann6e 1732, p. 444.
9 Ibid., anne*e 1737 (Paris, 1740), "Me*moires," p. 64; also ann& 1741 (Paris,
1744), p. 36.
10 Ibid., ann6e 1741, p. 282.
11 Ibid., ann6e 1744 (Paris, 1748), p. 406.
12 Ibid., ann6e 1745 (Paris, 1749), p. 369, 380.
13 Ibid., ann6e 1769 (Paris, 1772), p. 211.
" Ibid., anne*e 1774 (Paris, 1778), p. 103.
AGGREGATION 395
353. Terms in an aggregate placed in a vertical column. — The em-
ployment of a brace to indicate the sum of coefficients or factors
placed in a column was in vogue with Vieta (§176), Descartes, and
many other writers. Descartes in 1637 used a single brace,1 as in
— cc
or a vertical bar2 as in
— cc
zz
— ace
JLa*
,+*>* »o.
—\aacc
Wallis3 in 1685 puts the equation aaa-{-baa-\-cca = ddd, where a is the
unknown, also in the form
1
aaa
+ b
aa
+ CC
1
Sometimes terms containing the same power of x were written in a
column without indicating the common factor or the use of symbols of
aggregation; thus, John Wallis4 writes in 1685,
aaa + baa + bca = + bed
+caa— bda
— daa—cda
Giovanni Poleni5 writes in 1729,
y* + xxy4 — 2ax3yy + aax* = 0
— 2axy4 + aaxxyy
- aay4
The use of braces for the combination of terms arranged in col-
umns has passed away, except perhaps in recording the most unusual
algebraic expressions. The tendency has been, whenever possible, to
discourage symbolism spreading out vertically as well as horizontally.
Modern printing encourages progression line by line.
354. Marking binomial coefficients. — In the writing of the factors
in binomial coefficients and in factorial expressions much diversity of
practice prevailed during the eighteenth century, on the matter of
1 Descartes, (Euvres (e*d. Acfain et Tannery), Vol. VI, p. 450.
2 Ibid.
3 John Wallis, Treatise of Algebra (London, 1685), p. 160.
4 John Wallis, op. ciL, p. 153.
6 Joannis Poleni, Epistolarvm mathematicarvm fascicvlvs (Padua, 1729) (no
pagination).
396 A HISTORY OF MATHEMATICAL NOTATIONS
the priority of operations indicated by + and — , over the operations
of multiplication marked by • and X. Inn.n— 1-w — 2 or nXn — IX
n— 2, or n,n— 1, n—2, it was understood very generally that the sub-
tractions are performed first, the multiplications later, a practice con-
trary to that ordinarily followed at that time. In other words, these
expressions meant n(n— l)(n— 2). Other writers used parentheses or
vinculums, which removed all inconsistency and ambiguity. Nothing
was explicitly set forth by early writers which would attach different
meanings to nn and n-n or nXn. And yet, n«n— l*n— 2 was not the
same as nn— In— 2. Consecutive dots or crosses tacitly conveyed the
idea that what lies between two of them must be aggregated as if it
were inclosed in a parenthesis. Some looseness in notation occurs
even before general binomial coefficients were introduced. Isaac
Barrow1 wrote "L-MX :R+S" for (L-M)(fl+S), where the colon
designated aggregation, but it was not clear that L— M, as well as
R+S, were to be aggregated. In a manuscript of Leibniz2 one finds
the number of combinations of n things, taken k at a time, given in the
form
-2, etc., n — k+1
This diversity in notation continued from the seventeenth down
into the nineteenth century. Thus, Major Edward Thornycroft
(1704)3 writes mXrn — iXm— 2Xm— 3, etc. A writer4 in the Acta
eruditorum gives the expression n,n—l. Another writer5 gives
(n,n-"l,n-2)^ Leibniz>6 notation, as described in 1710 (§ 198), con-
Z,6
tains e*e — l*e — 2 for e(e—l)(e—2). Johann Bernoulli7 writes n«n —
l»w— 2. This same notation is used by Jakob (James) Bernoulli8 in a
1 Isaac Barrow, Lectiones mathematical, Lect. XXV, Probl. VII. See also
Probl. VIII.
2D. Mahnke, Bibliotheca mathematica (3d ser.), Vol. XIII (1912-13), p. 35.
See also Leibnizens Mathematische Schriften, Vol. VII (1863), p. 101.
3 E. Thornycroft in Philosophical Transactions, Vol. XXIV (London, 1704-5),
p. 1963.
*Acto eruditorum (Leipzig, 1708), p. 269.
*Ibid., Suppl., Tome IV (1711), p. 160. t
6 M iscellanea Berolinensia (Berlin, 1710), p. 161.
7 Johann Bernoulli in Acta eruditorum (1712), p. 276.
8 Jakob Bernoulli, Ars Conjectandi (Basel, 1713), p. 99.
AGGREGATION 397
posthumous publication, by F. Nicole1 who uses x+n»x+2n>x+3n>
etc., by Stirling2 in 1730, by Cramer3 who writes in a letter to J.
Stirling a^a+6«a+26, by Nicolaus Bernoulli4 in a letter to Stirling
r»r+&«r+2&. ... by Daniel Bernoulli5 Z— 1-Z — 2, by Lambert6 4m— !•
4m— 2, and by Konig7 n*n — 5«n— 6«n— 7. Euler8 in 1764 employs in
the same article two notations: one, n— 5«n — 6«n — 7; the other,
w(n — l)(n-2). Condorcet9 has n+2Xw+l. Hindcnburg10 of Got-
tingen uses round parentheses and brackets, nevertheless he writes
binomial factors thus, m-m—l>m — 2 . . . .m — s+l. Segner11 and
Ferroni12 write n *n — 1 -n — 2. Cossali13 writes 4X — 2= — 8. As late as
1811 A. M. Legendre14 has w-n— 1-n — 2 .... 1.
On the other hand, F. Nicole,15 who in 1717 avoided vinculums,
writes in 1723, a;«n+w-x+2n, etc. Stirling16 in 1730 adopts z—\ *z — 2.
De Moivre17 in 1730 likewise writes m—pXm — qX m — s, etc. Similar-
ly, Dodson,18 n*n-~ l»n— 2, and the Frenchman F. de Lalande,19
1 Nicole in Histoire de I'acad6mie r. des sciences, annec 1717 (Paris, 1719),
"Memoires," p. 9.
2 J. Stirling, Methodus dijjerentialis (London, 1730), p. 9.
3 Ch. Tweedic, James Stirling (Oxford, 1922), p. 121. 4 Op. tit., p. 144.
5 Daniel I. Bernoulli, "Notationes de aequationibus," Comment. Acad. Pelrop.,
Tome V (1738), p. 72.
6 J. H. Lambert, Observaliones in Ada Helvetica, Vol. III.
7 S. Konig, Histoire de V academic r. des sciences et des belles lettres, ann6e 1749
(Berlin, 1751), p. 189.
8 L. Euler, op. cit., annee 1764 (Berlin, 1760), p. 195, 225.
9 N. C. de Condorcet in Histoire de I'acadcmie r. des sciences, ann6e 1770 (Paris,
1773), p. 152.
10 Carl Fricdrich Hindenburg, Infinitinomii dignatum leges . . . . ac formulae
(Gottingen, 1779), p. 30.
11 J. A. de Segner, Cursus mathemalici, pars II (Halle, 1768), p. 190.
12 P. Ferroni, Magniludinum exponentialium .... theoria (Florence, 1782),
p. 29.
13 Pietro Cossali, Origine, trasporto in Italia ... dell1 algebra, Vol. I (Parma,
1797), p. 260.
14 A. M. Legendre, Exercices de calcul integral, Tome I (Paris, 1811), p. 277.
18 Histoire de Vacademie r. des sciences, ann6e 1723 (Paris, 1753), "Memoires,"
p. 21.
16 James Stirling, Methodus differ entialis (London, 1730), p. 6.
17 Abraham de Moivre, Miscellanea analytica de seriebus (London, 1730), p. 4.
18 James Dodson, Mathematical Repository, Vol. I (London, 1748), p. 238.
19 F. de Lalande in Histoire de I'academie r. des sciences, anne*e 1761 (Paris,
1763), p. 127.
308 A HISTORY OF MATHEMATICAL NOTATIONS
m«(m+l).(m+2). In Lagrange1 we encounter in 1772 the strictly
modern form (m+l)(m+2)(m+3), . . . . , in Laplace2 in 1778 the
form (t-l).(i-2) ____ (i-r+1).
The omission of parentheses unnecessarily aggravates the inter-
pretation of elementary algebraic expressions, such as are given by
Kirkman,3 viz., -3 = 3X-1 for -3 = 3X(-1), — wX—n for
(-w)(-n).
355. Special uses of parentheses. — A use of round parentheses and
brackets which is not strictly for the designation of aggregation is
found in Cramer4 and some of his followers. Cramer in 1750 writes
two equations involving the variables x and y thus:
A ____ x'-[-l]x»-l+[l*\x*-*-[l*]x»-*+ &c
B ____ (0)z°+(lK+(2)z2+(3)z3+ &c .....
where 1, I2, I3, . . . . , within the brackets of equation A do not mean
powers of unity, but the coefficients of x, which are rational functions
of y. The figures 0, 1, 2, 3, in B are likewise coefficients of x and func-
tions of y. In the further use of this notation, (02) is made to repre-
sent the product of (0) and (2); (30) the product of (3) and (0), etc.
Cramer's notation is used in Italy by Cossali5 in 1799.
Special uses of parentheses occur in more recent time. Thus
W. F. Sheppard8 in 1912 writes
(n,r) for n(w-l) ____ (n-r+l)/r!
[n,r] for n(n+l) ____ (n+r-l)/r!
for (n-s)(w
356. A star to mark the absence of terms. — We find it convenient
to discuss this topic at this time. Ren£ Descartes, in La Geometrie
(1637), arranges the terms of an algebraic equation according to the
descending order of the powers of the unknown quantity x, yy or z.
If any power of the unknown below the highest in the equation is
1 J. Lagrange in ibid., ann6e 1772, Part I (Paris, 1775), "M6moires," p. 523.
2 P. S. Laplace in ibid., ann6e 1778 (Paris, 1781), p. 237.
3 T. P. Kirkman, First Mnemonical Lessons in Geometry, Algebra and Trigonom-
etry (London, 1852), p. 8, 9.
< Gabriel Cramer, Analyse des Lignes cowrbes algebriqu&s (Geneva,* 1750),
p. 660.
5 Pietro Cossali, op. tit., Vol. II (Parma, 1799), p. 41.
6W. F. Sheppard in Fifth International Mathematical Congress, Vol. II, p. 355.
AGGREGATION 399
lacking, that fact is indicated by a *, placed where the term would
have been. Thus, Descartes writes z6— a46x = 0 in this manner:1
e * * *
He does not explain why there was need of inserting these stars in the
places of the missing terms. But such a need appears to have been
felt by him and many other mathematicians of the seventeenth and
eighteenth centuries. Not only were the stars retained in later edi-
tions of La Geometric, but they were used by some but not all of the
leading mathematicians, as well as by many compilers of textbooks.
Kinckhuysen2 writes "x? * * * * — &»0." Prestet3 in 1675 writes
a3**+63, and retains the * in 1689. The star is used by Baker,4 Varig-
non,5 John Bernoulli,6 Alexander,7 A. de Graaf,8 E. Hallcy.9 Fr. van
Schooten used it not only in his various Latin editions of Descartes7
Geometry, but also in 1646 in his Conic Sections,10 where he writes
z*»*-pz+qforz*= -pz+q. InW.Whiston's11 1707 edition of I. New-
ton's Universal Arithmetick one reads aa* — bb and the remark ". . . .
locis vacuis substituitur nota * ." Raphson's English 1728 edition of
the same work also uses the *. Jones12 uses * in 1706, Reyneau13 in
1708; Simpson14 employs it in 1737 and Waring15 in 1762. De Lagny16
1 Reno Descartes, La geometric (Leyden, 1637) ; (Euvres de Descartes (e*d.
Adam et Tannery), Vol. VI (1903), p. 483.
2 Gerard Kinckhuysen, Algebra ofte Stel-Konst (Haarlem, 1061), p. 59.
3 Elemens des mathematiques (Paris, 1675), Epttre, by J. P.[rostct], p. 23.
Nouvcaux elemens des Mathematiques, par Jean Prestet (Paris, 1689), Vol. II,
p. 450.
4 Thomas Baker, Geometrical Key (London, 1684), p. 13.
5 Journal des Sgavans, ann6e 1687, Vol. XV (Amsterdam, 1688), p. 459. The
star appears in many other places of this Journal.
6 John Bernoulli in Ada eruditorum (1688), p. 324. The symbol appears often
in this journal.
7 John Alexander, Synopsis Algebraica ... (Londini, 1693), p. 203.
8 Abraham de Graaf, De Geheele Mathesis (Amsterdam, 1694), p. 259.
9 E. Halley in Philosophical Transactions, Vol. XIX (London, 1695-97), p. 61.
10 Francisci a Schooten, De organica conicarum sectionum (Leyden, 1646), p. 91.
11 Arithmetica universalis (Cambridge, 1707), p. 29.
12 W. Jones, Synopsis palmariorum matheseos (London, 1706), p. 178.
"Charles Reyneau, Analyse demontree, Vol. I (Paris, 1708), p. 13, 89.
14 Thomas Simpson, New Treatise of Fluxions (London, 1737), p. 208.
> w Edward Waring, Miscellanea Analytica (1762), p. 37.
16 Memoires de Vacademie r. d. sciences. Depuis 1666 jusqu'a 1699, Vol. XI
(Paris, 1733), p. 241, 243, 250.
400 A HISTORY OF MATHEMATICAL NOTATIONS
employs it in 1733, De Gua1 in 1741, MacLaurin2 in his Algebra, and
Fenn3 in his Arithmetic. But with the close of the eighteenth century
the feeling that this notation was necessary for the quick understand-
ing of elementary algebraic polynomials passed away. In more ad-
vanced fields the star is sometimes encountered in more recent authors.
Thus, in the treatment of elliptic functions, Weicrstrass4 used it to
mark the absence of a term in an infinite series, as do also Greenhill5
and Fricke.6
1 Histoire de I' academic r. d. sciences, anne*e 1741 ( Paris, 1744), p. 476.
2 Colin Maclaurin, Treatise of Algebra (2d 6d.; London, 1756), p. 277.
3 Joseph Fcnn, Universal Arithmetic (Dublin, 1772), p. 33.
4 II. A. Schwarz, Formeln und Lehrsdtze .... nach Vorlesungen des Weier-
sirass (Gottingen, 1885), p. 10, 11.
5 A. G. Greenhill, Elliptic Functions (1892), p. 202, 204.
6R. Fricke, Encyklopadie d. Math. Wissenschaften, Vol. II2 (Leipzig, 1913),
p. 269.
IV
SYMBOLS IN GEOMETRY
(ELEMENTARY PART)
A. ORDINARY ELEMENTARY GEOMETRY
357. The symbols sometimes used in geometry may be grouped
roughly under three heads: (1) pictographs or pictures representing
geometrical concepts, as A representing a triangle; (2) ideographs de-
signed especially for geometry, as ^> for "similar"; (3) symbols of
elementary algebra, like + and — .
Early use of pictographs. — The use of geometrical drawings goes
back at least to the time of Ahmes, but the employment of pictographs
in the place of words is first found in Heron's Dioptra. Heron (150
A.D.) wrote /s for triangle, -^ for parallel and parallelogram, also ~gl
for parallelogram, r~V for rectangle, Q for circle.1 Similarly, Pappus
(fourth century A.D.) writes O and O for circle, v and A for triangle,
L for right angle, Ji. or == for parallel, D for square.2 But these were
very exceptional uses not regularly adopted by the authors and occur
in few manuscripts only. They were not generally known and are not
encountered in other mathematical writers for about one thousand
years. Paul Tannery calls attention to the use of the symbol D in a
medieval manuscript to represent, not a square foot, but a cubic foot;
Tannery remarks that this is in accordance with the ancient practice
of the Romans.8 This use of the square is found in the Triparty of
Chuquet (§ 132) and in the arithmetic of De la Roche.
358. Geometric figures were used in astrology to indicate roughly
the relative positions of two heavenly bodies with respect to an ob-
server. Thus 6, cP, D, A, H< designated,4 respectively, conjunction,
1 Notices et extraits des manuscrits de la Bibliothkque imp&riale, Vol. XIX,
Part II (Paris, 1858), p. 173.
2 Pappi Alexandrini Cottectionis quae supersuni (ed. F. Hultsch), Vol. Ill,
Tome I (Berlin, 1878), p. 126-31.
8 Paul Tannery, M&moires scientifiques, Vol. V (Toulouse and Paris, 1922),
p. 73.
4 Kepler says: "Quot sunt igitur aspectus? Vetus astrologia agnoscit t ant urn
quinque: conjunctionem (<$), cum radii planetarum binorum in Terram de-
scendentes in unam conjunguntur lineam; quod eat veluti principium aspectuum
omnium. 2) Oppositionem (cP)» cum bini radii sunt ejusdem rectae partes, seu
401
402 A HISTORY OF MATHEMATICAL NOTATIONS
opposition, at right angles, at 120°, at 60°. These signs are repro-
duced in Christian Wolff's Mathematisches Lexicon (Leipzig, 1716),
page 188. The #, consisting of three bars crossing each other at 60°,
was used by the Babylonians to indicate degrees. Many of their war
carriages are pictured as possessing wheels with six spokes.1
359. In Plato of Tivoli's translation (middle of twelfth century)
of the Liber embadorum by Savasorda who was a Hebrew scholar, at
Barcelona, about 1100 A.D., one finds repeatedly the designations
abc, ab for arcs of circles.2 In 1555 the Italian Fr. Maurolycus8 employs
A, D, also %. for hexagon and / .* for pentagon, while in 1575 he also
used CU. About half a century later, in 1623, Metius in the Nether-
lands exhibits a fondness for pictographs and adopts not only t^, D,
but a circle with a horizontal diameter and small drawings represent-
ing a sphere, a cube, a tetrahedron, and an octohedron. The last four
were never considered seriously for general adoption, for the obvious
reason that they were too difficult to draw. In 1634, in France, H6ri-
gone's Cursus mathematicus (§ 189) exhibited an eruption of symbols,
both pictographs and arbitrary signs. Here is the sign < for angle,
the usual signs for triangle, square, rectangle, circle, also J for right
angle, the Heronic = for parallel, <^> for parallelogram, ^ for arc of
circle, f* for segment, — for straight line, _L for perpendicular, 5<
for pentagon, 6< for hexagon.
In England, William Oughtred introduced a vast array of char-
acters into mathematics (§§ 181-85); over forty of them were used in
symbolizing the tenth book of Euclid's Elements (§§ 183, 184), first
printed in the 1648 edition of his Clavis mathematicae. Of these sym-
bols only three were pictographs, namely, Q for rectangle, D for
square, A for triangle (§ 184). In the first edition of the Clavis (1631),
the Q alone occurs. In the Trigonometria (1657), he employed /. for
angle and £ for angles (§ 182), || for parallel occurs in Oughtred's
cum duae quartae partes circuli a binis radiis interceptae sunt, id est unus semi-
circulus. 3) Tetragonum seu quadratum (D), cum una quarta. 4) Trigonumseu
trinum ( A), cum una tertia seu duae sextae. 5) Hexagonum seu sextilem (>(<), cum
una sexta." See Kepler, Opera omnia (ed. Ch. Frisch), Vol. VI (1866), p. 490,
quoted from "Epitomes astronomiae" (1618).
1 C. Bezold, Ninive und Babylon (1903), p. 23, 54, 62, 124. See also J. Tropfke,
op. tit., Vol. I (2d ed., 1921), p. 38.
2 See M. Curtze in Bibliotheca mathematica (3d ser.), Vol. I (1900), p. 327, 328.
3 Frantisci Maurolyd Abbatis Messanensis Opuscula Mathematica (Venice,
1575), p. 107, 134. See also Francisco Maurolyco in Boncompagni's Bulletino, Vol.
IX, p. 67.
GEOMETRY 403
Opuscula mathematica hactenus inedita (1677), a posthumous work
(§ 184).
Kltigel1 mentions a cube O as a symbol attached to cubic meas-
ure, corresponding to the use of n in square measure.
Euclid in his Elements uses lines as symbols for magnitudes, in-
cluding numbers,2 a symbolism which imposed great limitations upon
arithmetic, for he does not add lines to squares, nor does he divide a
line by another line.
360. Signs for angles. — We have already seen that H6rigone
adopted < as the sign for angle in 1634. Unfortunately, in 1631,
Harriot's Artis analyticae praxis utilized this very symbol for "less
than." Harriot's > and < for "greater than" and "less than" were
so well chosen, while the sign for "angle" could be easily modified so as
to remove the ambiguity, that the change of the symbol for angle was
eventually adopted. But < for angle persisted in its appearance,
especially during the seventeenth and eighteenth centuries. We find
it in W. Leybourn,3 J. Kersey,4 E. Hatton,6 E. Stone,8 J. Hodgson,7
D'Alembert's Encyclopedie* Hall and Steven's Euclid* and Th.
Reye.10 John Caswell11 used the sign g to express "equiangular."
A popular modified sign for angle was Z , in which the lower stroke
is horizontal and usually somewhat heavier. We have encountered
this in Oughtred's Trigonometria (1657), Caswell,12 Dulaurens,13
!G. S. Kliigel, Math. Worterbuch, 1. Theil (Leipzig, 1803), art. "Bruch-
zeichen."
2 Sec, for instance, Euclid's Elements, Book V; see J. Gow, History of Greek
Mathematics (1884), p. 106.
3 William Leybourn, Panorganon: or a Universal Instrument (London, 1672),
p. 75.
4 John Kersey, Algebra (London, 1673), Book IV, p. 177.
6 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 287.
6 Edmund Stone, New Mathematical Dictionary (London, 1726; 2d ed., 1743),
art. "Character."
7 Jarncs Hodgson, A System of Mathematics, Vol. I (London, 1723), p. 10.
8 Encyclopedic ou Dictionnaire raissonne, etc. (Diderot), Vol. VI (Lausanne et
Berne, 1781), art. "Caractere."
9 H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and U (London,
1889), p. 10.
10 Theodor Reye, Die Geometric der Laqe (5th ed.; Leipzig, 1909), 1. Abteilung,
p. 83.
11 John Caswell, "Doctrine of Trigonometry," in Wallis' Algebra (1685).
12 John Caswell, "Trigonometry," in ibid.
18 Francisci Dulaurens, Specimina mathematica duobus libris comprehensa
(Paris, 1667), "Symbols."
404 A HISTORY OF MATHEMATICAL NOTATIONS
Jones,1 Emerson,2 Hutton,3 Fuss,4 Steenstra,6 Kliigel,6 Playfair,7
Kambly,8 Wentworth,9 Fiedler,10 Casey,11 Lieber and von Ltihmann,12
Byerly,13 Muller,14 Mehler,15 C, Smith,16 Beman and Smith,17 Layng,18
Hopkins,19 Robbins,20 the National Committee (in the U.S.A.).21
The plural "angles" is designated by Caswell Z Z ; by many others
thus, A. Caswell also writes ZZ.Z. for the "sum of two angles,"
and X£ £ for the "difference of two angles." From these quotations
it is evident that the sign Z for angle enjoyed wide popularity in
different countries. However, it had rivals.
361. Sometimes the same sign is inverted, thus 7 as in John
Ward.22 Sometimes it is placed so as to appear A, as in the Ladies
1 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 221.
2 [W. Emerson], Elements of Geometry (London, 1763), p. 4.
8 Charles Hutton, Mathematical and Philosophical Dictionary (1695), art.
"Characters.'1
4 Nicolas Fuss, Lemons de geometrie (St. Petersbourg, 1798), p. 38.
5 Pibo Steenstra, Grondbeginsels der Meetkunst (Leyden, 1779), p. 101.
6 G. S. Kliigel, Math. Worterbuch, fortgesetzt von C. B. Mollweide und J. A.
Grunert, 5. Theil (Leipzig, 1831), art. "Zeichen."
1 John Playfair, Elements of Geometry (Philadelphia, 1855), p. 114.
8L. Kambly, Die Elementar-Mathematik, 2. Theil: Planimetrie, 43. Aufl.
(Breslau, 1876).
9 G. A. Wentworth, Ekments of Geometry (Boston, 1881; Preface, 1878).
10 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 7.
11 John Casey, Sequel to the First Six Books of the Elements of Euclid (Dublin,
1886).
13 H. Lieber und F. von Liihmann, Geometrische Konstruktions-Aufgaben,
8. Aufl. (Berlin, 1887), p. 1.
13 W. E. Byerly's edition of Chauvenet's Geometry (Philadelphia, 1905), p. 44.
14 G. Muller, Zeichnende Geometric (Esslingen, 1889), p. 12.
15 F. G. Mehler, Hauptsdtze der Elementar Mathematik, 8. Aufl. (Berlin, 1894),
p. 4.
16 Charles Smith, Geometrical Conies (London, 1894).
17 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896),
p. 10.
18 A. E. Layng, Euclid's Elements of Geometry (London, 1890), p. 4.
19 G. Irving Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12.
20 E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16.
21 Report by the National Committee on Mathematical Requirements, under the
auspices of the Mathematical Association of America, Inc. (1923), p. 77.
22 John Ward, The Young Mathematiciana' Guide (9th ed.; London, 1752),
p. 301, 369.
GEOMETRY 405
Diary1 and in the writings of Reyer,2 Bolyai,8 and Ottoni.4 This posi-
tion is widely used in connection with one or three letters marking an
angle. Thus, the angle ABC is marked by L. N. M. Carnot5 ABC in
his Geometric de position (1803); in the Penny Cyclopedia( 1839), arti-
cle "Sign," there is given A B; Binet,6 Mobius,7 and Favaro8 wrote
ab as the angle formed by two straight lines a and b; Favaro wrote
also PDC. The notation a b is used by Stob and Gmeiner,9 so that
/\ /\ X\ /\ X\
a b=—b a; Nixon10 adopted A, also ABC; the designation APM
is found in Enriques,11 Borel,12 and Durrell.13
362. Some authors, especially German, adopted the sign <£ for
angle. It is used by Spitz,14 Fiedler,15 Halsted,16 Milinowski,17 Meyer,13
1 Leybourne's Ladies Diary, Vol. IV, p. 273.
2 Samuel Reyhers .... Euclides, dessen VI. erste Backer auf sonderbare Art
mil algebraischen Zeichen, also eingerichtet, sind, dass man derselben Beweise auch in
anderen Sprachen gebrauchen kann (Kiel, 1698).
8 Wolfgang! Bolyai de Bolya, Tentamen (2d ed.), Tome II (Budapestini,
1904; 1st ed., 1832), p. 361.
4 C. B. Ottoni, Elementos de Geometria e Trigonometria (4th ed.; Rio de Janeiro,
1874), p. 67.
5 See Ch. Babbage, "On the Influence of Signs in Mathematical Reasoning,"
Transactions Cambridge Philos. Society, Vol. II (1827), p. 372.
« J. P. Binet in Journal de Vecole polyt., Vol. IX, Cahier 16 (Paris, 1813), p. 303.
7 A. F. Mobius, Gesammelte Werke, Vol. I (Leipzig, 1885), "Barycyentrischer
Calcul, 1827," p. 618.
8 A. Favaro, Lemons de Statique graphique, trad, par Paul Terrier, 1. Partie
(Paris, 1879), p. 51, 75.
9 O. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902),
p. 329, 330.
10 R. C. J. Nixon, Euclid Revised (3d ed.; Oxford, 1899), p. 9.
11 Fedcrigo Enriques, Questioni riguardanti la geometria elementare (Bologna,
1900), p. 67.
12 Emile Borel, Algebre (2d cycle; Paris, 1913), p. 367.
13 Clement V. Durell, Modern Geometry; The Straight Line and Circle (London,
1920), p. 7, 21, etc.
14 Carl Spitz, Lehrbuch der ebenen Geometric (Leipzig und Heidelberg, 1862),
p. 11.
15 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 7.
16 George Bruce Halsted, Mensuration (Boston, 1881), p. 28; Elementary Syn-
thetic Geometry (New York, 1892), p. vii.
17 A. Milinowski, Elcm.-Synth. Geom. der Kegelschnitte (Leipzig, 1883), p. 3.
18 Fricdrich Meyer, Dritter Cursus der Planimetrie (Halle, a/S, 1885), p. 81.
406 A HISTORY OF MATHEMATICAL NOTATIONS
Fialkowski,1 Henrici and Treutlein,2 Bruckner ,3 Doehlerfiann,4
Schur,5 Bernhard,6 Auerbach and Walsh,7 Mangoldt.8
If our quotations arc representative, then this notation for angle
finds its adherents in Germany and the United States. A slight modi-
fication of this sign is found in Byrne9 <0.
Among sporadic representations of angles are the following: The
capital letter10 L, the capital letter11 V, or that letter inverted,12 A, the
inverted capital letter13 \/J the perpendicular lines14 _] or L, pq the
angle made by the lines15 p and #, (a6) the angle between the rays,16 a
and b, ab the angle between the lines17 a and b, or (u, v) the angle18
formed by u and v.
363. Passing now to the designation of special angles we find ^
used to designate an oblique angle.19 The use of a pictograph for the
designation of right angles was more frequent in former years than
now and occurred mainly in English texts. The two perpendicular
lines L: to designate "right angle" are found in Reyher;20 he lets
1 N. Fialkowski, Praklische Geometric (Wien, 1892), p. 15.
2 J. Henrici und P. Treutlein, Lehrbuch der Elementar-Geometrie, 1. Teil, 3.
Aufl. (Leipzig, 1897), p. 11.
3 Max Bruckner, Vielecke und Vielfache-Theorie und Geschichte (Leipzig, 1900),
p. 125.
4 Karl Doehlemann, Projektive Geometric, 3. Aufl. (Leipzig, 1905), p. 133.
5 F. Schur, Grundlagen der Geometric (Leipzig und Berlin, 1909), p. 79.
9 Max Bernhard, Darstellende Geometric (Stuttgart, 1909), p. 267.
7 Matilda Auerbach and Charles B. Walsh, Plane Geometry (Philadelphia,
[1920]), p. vii.
8 Hans V. Mangoldt, Einfiihrung in die hohere Mathematik, Vol. I (Leipzig,
1923), p. 190.
9 Oliver Byrne, Elements of Euclid (London, 1847), p. xxviii.
10 John Wilson, Trigonometry (Edinburgh, 1714), " Characters Explained."
11 A. Saverien, Dictionnaire de math, et phys. (Paris, 1753), "Caractere."
12 W. Bolyai, Tentamen (2d ed.), Vol. I (1897), p. xi.
13 Joseph Fenn, Euclid (Dublin, 1769), p. 12; J. D. Blassiere, Principes de
geometric elementaire (The Hague, 1782), p. 16.
14 H. N. Robinson, Geometry (New York, 1860), p. 18; ibid. (15th ed., New
York), p. 14.
16 Charlotte Angas Scott, Modern Analytical Geometry (London, 1894), p. 253.
16 Heinrich Schroter, Theorie der Kegelschnitte (2d ed; Leipzig, 1876), p. 5.
17 J. L. S. Hatton, Principles of Projective Geometry (Cambridge, 1913), p. 9.
18 G. Peano, Formulaire mathematique (Turin, 1903), p. 266.
19 W. N. Bush and John B. Clarke, Elements of Geometry (New York, [1905]).
20 Samuel Reyhers Euclides (Kiel, 1698).
GEOMETRY 407
MIL stand for "angle A is a right angle," a symbolism which could
be employed in any language. The vertical bar stands for equality
(§ 263). The same idea is involved in the signs a t 6, i.e., "angle a is
equal to angle 6." The sign L for right angle is found in Jones,1
Hatton,2 Savericn,3 Fenn,4 and Steenstra.5 Kersey6 uses the sign -i. ,
Byrne7 Q>. Mach8 marks right angles i. The Frenchman Hdrigone9
used the sign J, the Englishman Dupius10 "1 for right angle.
James Mills Peirce,11 in an article on the notation of angles, uses
11 Greek letters to denote the directions of lines, without reference to
their length. Thus if p denotes the axis in a system of polar co-ordi-
nates, the polar angle will be p." Accordingly, = — #•
More common among more recent American and some English
writers is the designation "rt. Z" for right angle. It is found in
G. A. Wentworth,12 Byerly's Chauvenet,™ Hall and Stevens,14 Beman
and Smith,15 Hopkins,16 Robbins,17 and others.
Some writers use instead of pictographs of angles abbreviations
of the word. Thus Legendre18 sometimes writes "Angl. ACB"\
1 William Jones, Synopsis palrnariorum matheseos (London, 1706), p. 221.
2 Edward Hatton, An Intire System of Arithmetik (London, 1721), p. 287.
3 A. Saverien, Dictionnaire, "Caractere."
4 Joseph Fenn, Euclid (Dublin, 1769), p. 12.
5 Pibo Steenstra, Grondbeginsels der Meetkunst (Leyden, 1779), p. 101.
6 John Kersey, Algebra (London, 1673), Book IV, p. 177.
7 Oliver Byrne, The Elements of Euclid (London, 1847), p. xxviii.
8E. Mach, Space and Geometry (trans. T. J. McCormack, 1906), p. 122.
9 P. Herigone, Cursus mathematicus (Paris, 1634), Vol. I, "Explicatio no-
tarum."
10 N. F. Dupius, Elementary Synthetic Geometry (London, 1889), p. 19.
11 J. D. Runkle's Mathematical Monthly, Vol. I, No. 5 (February, 1859), p. 168,
169.
12 G. A. Wentworth, Elements of Plane and Solid Geometry (3d ed. ; Boston,
1882), p. 14.
13 W. E. Byerly's edition of ChauveneCs Geometry (1887).
14 H. S. Hall and F. II. Stevens, Euclid 's Elements, Parts I and II (London,
1889), p. It).
15 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896), p.
10.
16 G. I. Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12.
17 E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16.
18 A. M. Legendre, Elements de Geomelrie (Paris, 1794), p. 42.
408 A HISTORY OF MATHEMATICAL NOTATIONS
A. von Frank,1 "Wkl," the abbreviation for Winkel, as in "Wkl
DOQ."
The advent of non-Euclidean geometry brought Lobachevski's
notation n (p) for angle of parallelism.3
The sign -Y- to signify equality of the angles, and _L to signify
the equality of the sides of a figure, are mentioned in the article
"Caractere" by D'Alembert in Diderot* Encyclopedic of 1754 and of
1781s and in the Italian translation of the mathematical part (1800) ;
also in Rees's Cyclopaedia (London, 1819), article "Characters," and
in E. Stone's New Mathematical Dictionary (London, 1726), article
"Characters," but Stone defines _Y~ as signifying "equiangular or
similar." The symbol is given also by a Spanish writer as signifying
angulos iguales.* The sign — ° to signify "equal number of degrees"
is found in Palmer and Taylor's Geometry* but failed to be recom-
mended as a desirable symbol in elementary geometry by the Na-
tional Committee on Mathematical Requirements (1923), in their
Report, page 79.
Halsted suggested the sign «£ for spherical angle and also the
letter 12 to represent a "steregon," the unit of solid angle.6
364. Signs for "perpendicular." — The ordinary sign to indicate
that one line is perpendicular to another, _L, is given by H6rigone7 in
1634 and 1644. Another Frenchman, Dulaurens,8 used it in 1667. In
1673 Kersey9 in England employed it. The inverted capital letter j,
was used for this purpose by Caswell,10 Jones,11 Wilson,12 Saverien,18
1 A. von Frank in Archiv der Mathematik und Physik von J, A. Grunert (2d
ser.), Vol. XI (Leipzig, 1892), p. 198.
2 George Bruce Halsted, N. Lobatschewsky, Theory of Parallels (Austin, 1891),
p. 13.
3 Encydopedie au Dictionnaire raisonne des sciences , ... by Diderot, Vol. VI
(Lausanne et Berne, 1781), art. "Caractere."
4 Antonio Serra y Oliveres, Manual de la Tipografia Espanola (Madrid, 1852),
p. 70.
6 C. I. Palmer and D. P. Taylor, Plane Geometry (1915), p. 16.
6 G. B. Halsted, Mensuration (Boston, 1881), p. 28.
7 Pierre Herigone, Curvus mrthematicus, Vol. I (Paris, 1634), "Explicatio
notarum."
8 F. Dulaurens, Specimina mathematica dudbus libris comprehewa (Paris,
1667), "Symbols.0
9 John Kersey, Algebra (London, 1673), Book IV, p. 177.
10 J. Caswell's Trigonometry in J. Wallifl' Algebra (1685).
11 W. Jones, op. tit., p. 253.
12 J. Wilson, Trigonometry (Edinburgh, 1714), "Characters Explained."
13 A. Saverien, Lhctionnaire, "Caractere."
GEOMETRY 409
and Mauduit.1 Emerson2 has the vertical bar extremely short, -*-.
In the nineteenth century the symbol was adopted by all writers using
pictographs in geometry. Sometimes _k was used for ' 'perpendiculars. ' '
Thomas Baker3 adopted the symbol ^ for perpendicular.
365. Signs for triangle, square, rectangle, parallelogram. — The signs
A, D, CH or [), O are among the most widely used pictographs. We
have already referred to their occurrence down to the time of Hdrigone
and Oughtred (§184). The O for parallelogram is of rare occurrence
in geometries preceding the last quarter of the nineteenth century,
while the A, D, and CD occur in van Schooten,4 Dulaurens,5 Kersey,6
Jones,7 and Saverien.8 Some authors use only two of the three. A
rather curious occurrence is the Hebrew letter "mem," C2, to repre-
sent a rectangle; it is found in van Schooten,9 Jones,10 John Alexander,11
John I Bernoulli,12 Ronayne,13 Kliigel's Worterbuch™ and De Graaf.15
Newton,16 in an early manuscript tract on fluxions (October, 1666),
indicates the area or fluent of a curve by prefixing a rectangle to the
"~"* /
ordinate (§ 622), thus D — r-r- — , where x is the abscissa, and the
ao~iXx
fraction is the ordinate.
After about 1880 American and English school geometries came
to employ less frequently the sign a for rectangle and to introduce
more often the sign O for parallelogram. Among such authors are
1 A. R. Mauduit, Inleiding tot de Kleegel-Sneeden (The Hague, 1763), "Sym-
bols."
2 [W. Emerson], Elements of Geometry (London, 1763).
3 Thomas Baker, Geometrical Key (London, 1684), list of symbols.
4 Fr. van Schooten, Exercitationvm mathematicorvm liber primm (Leyden,
1657).
6 F. Dulaurens, loc. tit., "Symbols."
6 J. Kersey, Algebra (1673).
7 W. Jones, op. til., p. 225, 238.
8 Saverien, loc. tit.
9 Franciscus van Schooten, op. cit. (Leyden, 1657), p. 67.
10 W. Jones, op. til., p. 253.
11 Synopsis Algebraica, opus poslhumum lohannis Alexandri (London, 1693),
p. 67.
12 John Bernoulli in Ada eruditorum (1689), p. 586; ibid. (1692), p. 31.
18 Philip Ronayne, Treatise of Algebra (London, 1727), p. 3.
M J. G. Klugel, Math. Worterbuch, 5, Theil (Leipzig, 1831), "Zeichen."
16 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 81.
18 S. P. Rigaud, Historical Essay on .... Newton's Printipia (Oxford, 1838),
Appendix, p. 23.
410 A HISTORY OF MATHEMATICAL NOTATIONS
Halsted,1 Wentworth,2 Byerly,8 in his edition of Chauvenet, Beman
and Smith,4 Layng,5 Nixon,6 Hopkins,7 Robbins,8 and Lyman.9 Only
seldom do both O and £7 appear in the same text. Halsted10 denotes a
parallelogram by \\g'm.
Special symbols for right and oblique spherical triangles, as used
by Jean Bernoulli in trigonometry, are given in Volume II, § 524.
366. The square as an operator. — The use of the sign D to mark
the operation of squaring has a long history, but never became popu-
lar. Thus N. Tartaglia11 in 1560 denotes the square on a line tc in the
expression "il D de. tc." Cataldi12 uses a black square to indicate the
square of a number. Thus, he speaks of 8J-J-, "il suo | e 75 \\ -fj}."
Stampioen13 in 1640 likewise marks the square on BC by the
"D BC." Caramvel14 writes "D25. est Quadratum Numeri 25. hocest,
625."
A. de Graaf15 in 1672 indicates the square of a binomial thus:
I/a ±|/6, "zijn D is a+b±2i/ab." Johann I Bernoulli16 wrote
3 D Vax+xx for 3|/(ax+x2)2. Jakob Bernoulli in 169017 designated
1 G. B. Halsted, Elem. Treatise on Mensuration (Boston, 1881), p. 28.
2 G. A. Wentworth, Elements of Plane and Solid Geometry (3d ed. ; Boston,
1882), p. 14 (1st ed., 1878).
3 W. E. Byerly 's edition of Chauvenet' s Geometry (1887), p. 44.
4 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896),
p. 10.
5 A. E. Layng, Euclid's Elements of Geometry (London, 1890), p. 4.
8 R. C. J. Nixon, Euclid Revised (3d ed., Oxford, 1899), p. 6.
7 G. J. Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12.
8E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16.
9E. A. Lyman, Plane and Solid Geometry (New York, 1908), p. 18.
10 G. B. Halsted, Rational Geometry (New York, 1904), p. viii.
11 N. Tartaglia, La Quinta parte del general trattato de nvmeri et misvre (Venice,
1560), fols.82A£ and 83A.
12 Trattato del Modo Brevissimo de trouare la Radice quadra delli numeri, ....
Di Pietro Antonio Cataldi (Bologna, 1613), p. 111.
13 J. Stampioen, Wis-Konstich ende Reden-maetich Bewys ('S Graven-Hage,
1640), p. 42.
14 Joannis Caramvelis mathesis biceps, vetus, et nova (1670), p. 131.
15 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 32.
16 Johannis I Bernoulli, Lectiones de calculo differentialium .... von Paul
Schafheitlin, Separatabdruck aus den Verhandlungen der Naturforschenden Ge-
sellschaft in Basel, Vol. XXXIV (1922).
17 Jakob Bernoulli in Ada erudiiorum (1690), p. 223.
GEOMETRY 411
the square of £ by QOf , but in his collected writings1 it is given in the
modern form (|)2. Sometimes a rectangle, or the Hebrew letter
"mem," is used to signify the product of two polynomials.2
367. Sign for circle. — Although a small image of a circle to take
the place of the word was used in Greek time by Heron and Pappus,
the introduction of the symbol was slow. H6rigone used O, but
Oughtred did not. One finds O in John Kersey,3 John Caswell,4
John Ward,5 P. Steenstra,6 J. D. Blasstere,7 W. Bolyai,8 and in the
writers of the last half -century who introduced the sign O for paral-
lelogram. Occasionally the central dot is omitted and the symbol O
is used, as in the writings of Reyher9 and Saverien. Others, Fenn for
instance, give both O and O, the first to signify circumference, the
second circle (area). Caswell10 indicates the perimeter by O. Metius11
in 1623 draws the circle and a horizontal diameter to signify circulus.
368. Signs for parallel lines. — Signs for parallel lines were used by
Heron and Pappus (§ 701); H£rigone used horizontal lines = (§ 189)
as did also Dulaurens12 and Reyher,13 but when Recorders sign of
equality won its way upon the Continent, vertical lines came to be
used for parallelism. We find || for "parallel" in Kersey,14 Caswell,
Jones,15 Wilson,16 Emerson,17 Kambly,18 and the writers of the last
1 Opera Jakob Bernoulli.?, Vol. I, p. 430, 431; see G. Enestrom, Bibliotheca
mathematica (3d ser.), Vol. IX (1908-9), p. 207.
2 See P. Herigone, Cursus mathematici (Paris, 1644), Vol. VI, p. 49.
3 John Kersey, Algebra (London, 1673), Book IV, p. 177.
4 John Caswell in Wallis' Treatise of Algebra, "Additions and Emendations/'
p. 166. For "circumference" Caswell used the small letter c.
5 J. Ward, The Young Mathematician's Guide (9th cd.; London, 1752), p. 301,
369.
6 P. Steenstra, Grondbeginsels der Meetkunst (Ley den, 1779), p. 281.
7 J. D. Blassiere, Principes de geometric 6lemcntaire (The Hague, 1723), p. 16.
8 W. Bolyai, Tentamen (2d ed.), Vol. II (1904), p. 361 (1st ed., 1832).
9 Samuel Reyher s, Euclides (Kiel, 1698), list of symbols.
10 John Caswell in WalhV Treatise of Algebra (1685), "Additions and Emenda-
tions," p. 166.
11 Adriano Metio, Praxis nova geometrica (1623), p. 44.
12 Fr. Dulaurens, Spedmina mathematica (Paris, 1667), "Symbols."
13 S. Reyher, op. cit. (1698), list of symbols.
14 John Kersey, Algebra (London, 1673), Book IV, p. 177.
16 W. Jones, Synopsis palmariorum matheseos (London, 1706).
16 John Wilson, Trigonometry (Edinburgh, 1714), characters explained.
17 [W. Emerson], Elements of Geometry (London, 1763), p. 4.
18 L. Kambly, Die Elementar-Mathematik, 2. Theil, Planimetrie, 43. Aufl.
(Breslau, 1876), p. 8.
412 A HISTORY OF MATHEMATICAL NOTATIONS
fifty years who have been already quoted in connection with other
pictographs. Before about 1875 it does not occur as often as do A,
D, a. Hall and Stevens1 use "par1 or ||" for parallel. Kambly2 men-
tions also the symbols -ff and 4= for parallel.
A few other symbols are found to designate parallel. Thus
John Bolyai in his Science Absolute of Space used |||. Karsten3 used
4£; he says: "Man pflege wohl das Zeichen :$ statt des Worts:
Parallel der Klirze wegen zu gebrauchen." This use of that symbol
occurs also in N. Fuss.4 Thomas Baker5 employed the sign »=* .
With Kambly # signifies rectangle. Haseler6 employs $ as
"the sign of parallelism of two lines or surfaces."
369. Sign for equal and parallel. — #is employed to indicate that
two lines are equal and parallel in Klugel's Worterbuch;1 it is used by
H. G. Grassmann,8 Lorey,9 Fiedler,10 Henrici and Treutlein.11
370. Signs for arcs of circles. — As early a writer as Plato of Tivoli
(§ 359) used ab to mark the arc ab of a circle. Ever since that time it
has occurred in geometric books, without being generally adopted. It is
found in H£rigone,12 in Reyher,13in Kambly,14 in Lieber and Luhmann.16
W. R. Hamilton18 designated by ^LF the arc "from F to L." These
1 H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London,
1889), p. 10.
2 L. Kambly, op. Git., 2. Theil, Planimetrie, 43. Aufl. (Breslau, 1876), p. 8.
3 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, 1. Theil (Greifswald,
1767), p. 254.
4 Nicolas Fuss, Lemons de geometric (St. Petersbourg, 1798), p. 13.
5 Thomas Baker, Geometrical Key (London, 1684), list of symbols.
0 J. F. Haseler, Anfangsgrunde der Arith., Alg., Geom. und Trig. (Lemgo),
Elementar-Geometrie (1777), p. 72.
7 G. S. Kliigel, Mathemalisches Worterbuch, fortgesetzt von C. B. Mollweide,
J. A. Grunert, 5. Theil (Leipzig, 1831), "Zeichen."
8H. G. Grassmann, Ausdehnungslehre von 1844 (Leipzig, 1878), p. 37; Werke
by F. Engel (Leipzig, 1894), p. 67.
9 Adolf Lorey, Lehrbuch der ebenen Geomelrie (Gera und Leipzig, 1868), p. 52.
10 Wilhelm Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 11.
11 J. Henrici und P. Treutlein, Lehrbuch der Elementar-Geometrie, 1. Teil, 3.
Aufl. (Leipzig, 1897), p. 37.
12 P. Herigone, op. ciL (Paris, 1644), Vol. I, "Explicatio notarum."
13 Samuel Reyhers, Euclides (Kiel, 1698), Vorrede.
14 L. Kambly, op. tit. (1876).
16 H. Lieber und F. von Luhmann, Geomelrische Konstructions-Aufgaben, 8.
Aufl. (Berlin, 1887), p. 1.
16 W. R. Hamilton in Cambridge & Dublin Math'l. Journal, Vol. I (1846), p. 262.
GEOMETRY 413
references indicate the use of ^ to designate arc in different countries.
In more recent years it has enjoyed some popularity in the United
States, as is shown by its use by the following authors: Halsted,1
Wells,2 Nichols,3 Hart and Feldman,4 and Smith.5 The National Com-
mittee on Mathematical Requirements, in its Report (1923), page 78,
is of the opinion that "the value of the symbol ^ in place of the short
word arc is doubtful/'
In 1755 John Landen6 used the sign (PQR) for the circular arc
which measures the angle PQR, the radius being unity.
371. Other pictographs. — We have already referred to Herigone's
use (§ 189) of 5< and 6< to represent pentagons and hexagons.
Reyher actually draws a pentagon. Occasionally one finds a half-
circle and a diameter ° to designate a segment, and a half-circle with-
out marking its center or drawing its diameter to designate an arc.
Reyher in his Euclid draws / \ for trapezoid.
Pictographs of solids are very rare. We have mentioned (§ 359)
those of Metius. Saverein7 draws |? A> MM > HI to stand, respec-
tively, for cube, pyramid, parallelepiped, rectangular parallelopiped,
but these signs hardly belong to the category of pictographs.
Dulaurens8 wrote IH for cube and GO for aequi quadrimensum. Joseph
Fenn9 draws a small figure of a parallelopiped to represent that solid,
as Metius had done. Halsted10 denotes symmetry by -I- .
Some authors of elementary geometries have used algebraic sym-
bols and no pictographs (for instance, Isaac Barrow, Karsten, Tac-
quet, Leslie, Legendre, Playfair, Chauvenet, B. Peirce, Todhunter),
but no author since the invention of symbolic algebra uses pictographs
without at the same time availing himself of algebraic characters.
372. Signs for similarity and congruence. — The designation of
"similar,'' "congruent/' "equivalent," has brought great diversity
of notation, and uniformity is not yet in sight.
Symbols for similarity and congruence were invented by Leibniz.
1G. B. Halsted, Mensuration (Boston, 1881).
2 Webster Wells, Elementary Geometry (Boston, 1886), p. 4.
3 E. H. Nichols, Elements of Constructional Geometry (New York, 1896).
4 C. A. Hart and D. D. Feldman, Plane Geometry (New York, [1911]), p. viii.
6 Eugene R. Smith, Plane Geometry (New York, 1909), p. 14.
6 John Landen, Mathematical Lucubrations (London, 17.55), Sec. Ill, p. 93.
7 A. Saverein, Dictionnaire, "Caractere."
8F. Dulaurens, op. cit. (Paris, 1667), "Symbols."
9 Joseph Fenn, Euclid's Elements of Geometry (Dublin, [ca. 1769]), p. 319.
10 G. B. Halsted, Rational Geometry (New York, 1904), p. viii.
414 A HISTORY OF MATHEMATICAL NOTATIONS
In Volume II, § 545, are cited symbols for "coincident" and "congru-
ent" which occur in manuscripts of 1679 and were later abandoned by
Leibniz. In the manuscript of his Characteristica Geometrica which was
not published by him, he says: "similitudinemitanotabimus: a~b.'n
The sign is the letter S (first letter in similis) placed horizontally.
Having no facsimile of the manuscript, we are dependent upon the
editor of Leibniz' manuscripts for the information that the sign in
question was ~ and not ^>. As the editor, C. I. Gerhardt, inter-
changed the two forms (as pointed out below) on another occasion,
we do not feel certain that the reproduction is accurate in the present
case. According to Gerhardt, Leibniz wrote in another manuscript
r^ for congruent. Leibniz7 own words are reported as follows: "ABC
~.CDA. Nam ~ inihi est signum similitudinis, et = aequalitatis,
unde congruentiae signum compono, quia quae simul et similia et
aequalia sunt, ea congrua sunt."2 In a third manuscript Leibniz
wrote |r^| for coincidence.
An anonymous article printed in the Miscellanea Berolinensia
(Berlin, 1710), under the heading of "Monitum de characteribus alge-
braicis," page 159, attributed to Leibniz and reprinted in his col-
lected mathematical works, describes the symbols of Leibniz; ^ for
similar and ^2. for congruent (§ 198). Note the change in form; in
the manuscript of 1679 Leibniz is reported to have adopted the form
~, in the printed article of 1710 the form given is ^. Both forms have
persisted in mathematical writings down to the present day. As re-
gards the editor Gerhardt, the disconcerting fact is that in 1863 he
reproduces the ^ of 1710 in the form3 ^.
The Leibnizian symbol ~ was early adopted by Christian von
Wolf; in 1716 he gave ~ for Aehnlichkeit,4 and in 1717 he wrote " = et
~" for "equal and similar."5 These publications of Wolf are the
earliest in which the sign ^ appears in print. In the eighteenth and
early part of the nineteenth century, the Leibnizian symbols for
"similar" and "congruent" were seldom used in Europe and not at all
in England and America. In England ^ or ^ usually expressed
"difference," as defined by Oughtred. In the eighteenth century the
signs for congruence occur much less frequently even than the signs
1 Printed in Leibnizens Math. Schrifien (ed. C. I. Gerhardt), Vol. V, p. 153.
2 Op. tit., p. 172.
3 Leibnizens Math. Schriften, Vol. VII (1863), p. 222.
4Chr. Wolffen, Math. Lexicon (Leipzig, 1716), "Signa."
5Chr. V. Wolff, Elementa Matheseos universalis (Halle, 1717), Vol. I, §236;
see Tropfke, op. tit., Vol. IV (2d ed., 1923), p. 20.
GEOMETRY 415
>r similar. We have seen that Leibniz' signs for congruence did not
se both lines occurring in the sign of equality = . Wolf was the first to
se explicitly ~ and = for congruence, but he did not combine the
NO into one symbolism. That combination appears in texts of the
itter part of the eighteenth century. While the ~ was more involved,
nee it contained one more line than the Leibnizian ^, it had the
dvantage of conveying more specifically the idea of congruence as
le superposition of the ideas expressed by ~ and =. The sign ^>
>r "similar" occurs in Camus' geometry,1 ^ for "similar" in A. R.
lauduit's conic sections2 and in Karsten,3 ^ in Blassiere's geometry,4
= for congruence in Haseler's5 and Reinhold's geometries,6 ^ for
milar in Diderot's Encyclopedic? and in Lorenz' geometry.8 In
ItigeFs Worterbuch? one reads, "^ with English and French authors
icans difference" ; "with German authors ^ is the sign of similarity" ;
Leibniz and Wolf have first used it." The signs ~ and ^ are used
y Mollweide;10 ~ by Steiner11 and Koppe;12 ^ is used by Prestel,13
= by Spitz;14 ~ and = are found in Lorey's geometry,15 Kambly's
1 C. E. L. Camus, Siemens de geometrie (nouvelle 6d.; Paris, 1755).
2 A. R. Mauduit, op. tit. (The Hague, 1763), "Symbols."
3 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, 1. Theil (1767),
348.
4 J. D. BlassieTe, Printipes de geometrie ttementaire (The Hague, 1787),
16.
8 J. F. Ilaseler, op. tit. (Lemgo, 1777), p. 37.
6 C. L. Reinhold, Arithmetica Forensis, 1. Theil (Ossnabriick, 1785), p. 361.
7 Diderot Encyclopedic on Dictionnaire raisone des sciences (1781; 1st ed.,
54), art. "Caractere" by D'Alembert. See also the Italian translation of the
athematical part of Diderot's Encyclopedic, the IHzionario enticlopedico delle
tiematiche (Padova, 1800), "Carattcrc."
8 J. F. Lorenz, Grundriss der Arithmetik und Geometrie (HelmstMt, 1798), p. 9.
9 G. S. Kliigel, Mathematisches Worterbuch, fortgesetzt von C. B. Mollweide,
A. Grunert, 5. Theil (Leipzig, 1831), art. "Zeichen."
10 Carl B. Mollweide, Euklid's Elemente (Halle, 1824).
11 Jacob Steiner, Geometrische Constructionen (1833); Ostwald's Klassiker, No.
,p.6.
12 Karl Koppe, Planimelrie . (Essen, 1852), p. 27.
u M. A. F. Prestel, Tabelarischer Grundriss der Experimental-physik (Emden,
56), No. 7.
14 Carl Spitz, Lehrbuch der ebenen Geometrie (Leipzig und Heidelberg, 1862),
41.
15 Adolf Lorey, Lehrbuch der ebenen Geometrie (Gera und Leipzig, 1868),
118.
416 A HISTORY OF MATHEMATICAL NOTATIONS
Planimetrie^ and texts by Frischauf2 and Max Simon.8 Lorey's book
contains also the sign ^ a few times. Peano4 uses ^ for "similar" also
in an arithmetical sense for classes. Perhaps the earliest use of ^ and
^ for "similar" and "congruent" in the United States are by G. A.
Hill5 and Halsted.6 The sign ^ for "similar" is adopted by Henrici
and Treutlein,7 ^ by Fiedler,8 ^ by Fialkowski,9 ^ by Beman and
Smith.10 In the twentieth century the signs entered geometries in the
United States with a rush: ^ for "congruent" were used by Busch
and Clarke;11 ^ by Meyers,12 ^ by Slaught and Lennes,13 ^ by Hart
and Feldman;14 ^ by Shutts/6 E. R. Smith,16 Wells and Hart,17 Long
and Brenke;18 ^ by Auerbach and Walsh.19
That symbols often experience difficulty in crossing geographic or
national boundaries is strikingly illustrated in the signs ~ and ^.
The signs never acquired a foothold in Great Britain. To be sure,
the symbol *-s was adopted at one time by a member of the University
1 L. Kambly, Die Elementar-Mathematik, 2. Thcil, Planimetrie, 43. Aufl.
(Breslau, 1876).
2 J. Frischauf, Absolute Geometric (Leipzig, 1876), p. 3.
3 Max Simon, Euclid (1901), p. 45.
4 G. Pcano, Formulaire de mathematiques (Turin, 1894), p. 135.
6 George A. Hill, Geometry for Beginners (Boston, 1880), p. 92, 177.
6 George Bruce Halsted, Mensuration (Boston, 1881), p. 28, 83.
7 J. Henrici und P. Treutlein, Elementar-Geometrie (Leipzig, 1882), p. 13, 40.
8 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 60.
9 N. Fialkowski, Praktische Geometrie (Wien, 1892), p. 15.
10 W. W. Bernan and D. E. Smith, Plane and Solid Geometry (Boston, 1896),
p. 20.
11 W. N. Busch and John B. Clarke, Elements of Geometry (New York,
1905]).
12 G. W. Meyers, Second-Year Mathematics for Secondary Schools (Chicago,
1910), p. 10.
13 II. E. Slaught and N. J. Lennes, Plane Geometry (Boston, 1910).
14 C. A. Hart and D. D. Feldman, Plane Geometry (New York, 1911),
p. viii.
16 G. C. Shutts, Plane and Solid Geometry [1912], p. 13.
16 Eugene R. Smith, Solid Geometry (New York, 1913).
17 W. Wells and W. W. Hart, Plane and Solid Geometry (Boston, [1915]),
p. x.
18 Edith Long and W. C. Brenke, Plane Geometry (New York, 1916), p. viii.
19 Matilda Auerbach and Charles Burton Walsh, Plane Geometry (Philadelphia,
[1920]), p. xi.
GEOMETRY 417
of Cambridge,1 to express "is similar to" in an edition of Euclid. The
book was set up in type, but later the sign was eliminated from all
parts, except one. In a footnote the student is told that "in writing
out the propositions in the Senate House, Cambridge, it will be ad-
visable not to make use of this symbol, but merely to write the word
short, thus, is simil." Moreover, in the Preface he is informed that
"more competent judges than the editor" advised that the symbol be
eliminated, and so it was, except in one or two instances where "it
was too late to make the alteration," the sheets having already been
printed. Of course, one reason for failure to adopt ^ for "similar" in
England lies in the fact that ^ was used there for "difference."
373. When the sides of the triangle ABC and A'B'C' are con-
sidered as being vectors, special symbols have been used by some
authors to designate different kinds of similarity. Thus, Stolz and
Grneiner2 employ ^ to mark that the similar triangles are uniformly
similar (einstimmig dhnlich), that is, the equal angles of the two tri-
angles are all measured clockwise, or all counter-clockwise; they em-
ploy ^ to mark that the two triangles are symmetrically similar,
that is, of two numerically equal angles, one is measured clockwise
and the other counter-clockwise.
The sign ^ has been used also for "is [or are] measured by," by
Alan Sanders;3 the sign 9= is used for "equals approximately," by
Hudson and Lipka.4 A. Pringsheim5 uses the symbolism av^abv to
express that , ^r^ — a.
1 v = + QO bv
374. The sign ^ for congruence was not without rivals during the
nineteenth century. Occasionally the sign =, first introduced by
Riernann6 to express identity, or non-Gaussian arithmetical congru-
ence of the type (a+6)2 = a2+2a6+62, is employed for the expression
of geometrical congruence. One finds == for congruent in W. Bolyai,7
1 Elements of Euclid .... from the Text of Dr. Simson. By a Member of the
University of Cambridge (London, 1827), p. 104.
2 O. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902),
p. 332.
3 Alan Sanders, Plane and Solid Geometry (New York, [1901]), p. 14.
4 R. G. Hudson and J. Lipka, Manual of Mathematics (New York, 1917), p. 68.
5 A. Pringsheim, Mathematische Annalen, Vol. XXXV (1890), p. 302; En-
cyclopedie des scien. Math., Tom. I, Vol. I (1904), p. 201, 202.
6 See L. Kronecker, Vorlesungen liber Zahlentheorie (Leipzig, 1901), p. 86;
G. F. B. Riemann, Elliptische Funktiown (Leipzig, 1899), p. 1, 6.
7 W. Bolyai, Tentamen (2d ed.), Tom. I (Budapest, 1897), p. xi.
418 A HISTORY OF MATHEMATICAL NOTATIONS
II . G. Grassrnann,1 Dupuis,2 Biulden,3 Veronese,4 Casey,5 Halsted,8
Baker,7 Betz and Webb,8 Young and Schwarz,9 McDougall.10 This
sign = for congruence finds its widest adoption in Great Britain at the
present time. Jordan11 employs it in analysis to express equivalence.
The idea of expressing similarity by the letter $ placed in a
horizontal position is extended by Callet, who uses w, O, Q, to
express "similar," "dissimilar," "similar or dissimilar/'12 Callet's
notation for "dissimilar" did not meet with general adoption even in
his own country.
The sign = has also other uses in geometry. It is used in the
Riemaimian sense of "identical to," not "congruent," by Busch and
Clarke,13 Meyers,14 E. R. Smith,15 Wells and Hart.16 The sign = or ><
is made to express "equivalent to" in the Geometry of Hopkins.17
The symbols ~ and ^ for "similar" have encountered some com-
petition with certain other symbols. Thus "similar" is marked |||
in the geometries of Budden18 and McDougall.
The relation "coincides with," which Leibniz had marked with
|~|, is expressed by =£ in White's Geometry.™ Cremona-0 denotes by
I H. G. Grassmann in C relic's Journal, Vol. XLII (1851), p. 193-203.
- N. F. Dupuis, Elementary Synthetic Geometry (London, 1899), p. 29.
3 E. Budden, Elementary Pure Geometry (London, 1904), p. 22.
4 Guiseppe Veronese, Elementi di Geomelria, Part I (3ded.; Verona, 1904), p. 11.
5 J. Casey, First Six Books of Euclid' 's Elements (7th ed.; Dublin, 1902).
6 G. B. Halsted, Rational Geometry (New York, 1904), p. vii.
7 Alfred Baker, Transactions of the Royal Society of Canada (2d ser., 1906-7),
Vol. XII, Sec. Ill, p. 120.
8 W. Betz and H. E. Webb, Plane Geometry (Boston, [1912]), p. 71.
a John W. Young and A. J. Schwartz, Plane Geometry (New York, [1905]).
10 A. II. McDougall, The Ontario High School Geometry (Toronto, 1914), p. 158.
II Camille Jordan, Cours (V analyse, Vol. II (1894), p. (314.
12 Francois Callet, Tables portaiires de loyarilhmes (Paris, 1795), p. 79. Taken
from Desire Andre, Notations ?nathcma(iques (Paris, 1909), p. 150.
13 W. N. Busch and John B. Clarke, Elements of Geometry (New York, [1905]).
14 G. W. Meyers, Second-Year Mathematics for /Secondary Schools (Chicago,
1910), p. 119.
16 Eugene R. Smith, Solid Geometry [1913].
18 W. Wells and W. W. Hart, Plane and Solid Geometry (Boston, [1905]), p. x.
17 Irving Hopkins, Manual of Plane Geometry (Boston, 1891), p. 10.
18 E. Budden, Elementary Pure Geometry (London, 1904), p. 22.
10 Emerson E. White, Elements of Geometry (New York City, .1895).
20 Luigi Cremona, Projectiue Geometry (trans. Oh. Leudesdorf; 2d ed.; Oxford,
1893), p. 1.
GEOMETRY 419
a.BC^A' that the point common to the plane a and the straight
line BC coincides with the point A'. Similarly, a German writer1 of
1851 indicates by a=6, A = B that the two points a and b or the two
straights A and B coincide (zusammenf alien) .
375. The sign =O for equivalence. — In many geometries congruent
figures are marked by the ordinary sign of equality, — . To distin-
guish between congruence of figures, expressed by =, and mere
equivalence of figures or equality of areas, a new symbol O came to
be used for "equivalent to" in the United States. The earliest appear-
ance of that sign known to us is in a geometry brought out by Charles
Davies2 in 1851. He says that the sign "denotes equivalency and is
read is equivalent to." The curved parts in the symbol, as iised by
Davies, are not semicircles, but semiellipses. The sign is given by
Davies and Peck,8 Benson,4 Wells,5 Went worth,6 McDonald,7
Macnie,8 Phillips and Fisher,9 Milne,10 McMahon,11 Durcll,12 Hart and
Feldman.13 It occurs also in the trigonometry of Anderegg and
Roe.14 The signs =0= and = for equivalence and equality (i.e. congru-
ence) are now giving way in the United States to = and ^ or ^ .
We have not seen this symbol for equivalence in any European
book. A symbol for equivalence, =£=, was employed by John Bolyai15
in cases like AB^CD, which meant ZCAB= Z.ACD. That the line
BN is parallel and equal to CP he indicated by the sign "BN\\*±CP"
1 Crelle's Journal, Vol. XLII (1851), p. 193-203.
2 Charles Davies, Elements of Geometry and Trigonometry from the Works of
A. M. Legendre (New York, 1851), p. 87.
3 Charles Davies and W. G. Peck, Mathematical Dictionary (New York, 1856),
art. "Equivalent."
4 Lawrence S. Benson, Geometry (New York, 1867), p. 14.
5 Webster Wells, Elements of Geometry (Boston, 1886), p. 4.
6G. A. Wentworth, Text-Book of Geometry (2d ed.; Boston, 1894; Preface,
1888), p. 16. The first edition did not use this symbol.
7 J. W. Macdonald, Principles of Plane Geometry (Boston, 1894), p. 6.
8 John Macnie, Elements of Geometry (ed. E. E. White; New York, 1895), p. 10.
9 A. W. Phillips and Irving Fisher, Elements of Geometry (New York, 1896), p. 1.
10 William J. Milne, Plane and Solid Geometry (New York, [1899]), p. 20.
11 James McMahon, Elementary Geometry (Plane) (New York, [1903]), p. 139.
12 Fletcher Durrell, Plane and Solid Geometry (New York, 1908), p. 8.
13 C. A. Hart and D. D. Feldman, Plane Geometry (New York, [1911]), p. viii.
14 F. Anderegg and E. D. Roe, Trigonometry (Boston, 1896), p. 3.
« W. Bolyai, Tentamen (2d ed.), Vol. II, Appendix by John Bolyai, list of
symbols. See also G. B. Halsted's translation of that Appendix (1896).
420 A HISTORY OF MATHEMATICAL NOTATIONS
376. Lettering of geometric figures* — Geometric figures are found in
the old Egyptian mathematical treatise, the Ahmes papyrus (1550
B.C. or older), but they are not marked by signs other than numerals
to indicate the dimensions of lines.
The designation of points, lines, and planes by a letter or by letters
was in vogue among the Greeks and has been traced back1 to Hip-
pocrates of Chios (about 440 B.C.).
The Greek custom of lettering geometric figures did not find imi-
tation in India, where numbers indicating size were written along the
sides. However, the Greek practice was adopted by the Arabs, later
still by Regiomontanus and other Europeans.2 Gerbert8 and his
pupils sometimes lettered their figures and at other times attached
Roman numerals to mark lengths and areas. The Greeks, as well as
the Arabs, Leonardo of Pisa, and Regiomontanus usually observed
the sequence of letters a, b, g, d, e, z, etc., omitting the letters c and/.
We have here the Greek-Arabic succession of letters of the alphabet,
instead of the Latin succession. Referring to Leonardo of Pisa's
Practica geometriae (1220) in which Latin letters are used with geo-
metric figures, Archibald says: "Further evidence that Leonardo's
work was of Greek- Arabic extraction can be found in the fact that, in
connection with the 113 figures, of the section On Divisions, of Leonar-
do's work, the lettering in only 58 contains the letters c or /; that is,
the Greek-Arabic succession a b g d e z . . . . is used almost as fre-
quently as the Latin abcdefg....; elimination of Latin letters
added to a Greek succession in a figure, for the purpose of numerical
examples (in which the work abounds), makes the balance equal. "4
Occasionally one encounters books in which geometric figures are
not lettered at all. Such a publication is ScheubePs edition of Euclid,5
in which numerical values are sometimes written alongside of lines as
in the Ahmes papyrus.
An oddity in the lettering of geometric figures is found in Ramus'
use6 of the vowels a, e, i, o, u, y and the employment of consonants
only when more than six letters are needed in a drawing.
1 M. Cantor, op. dt., Vol. I (3d ed., 1907), p. 205.
2 J. Tropfke, op. tit., Vol. IV (2d ed., 1923), p. 14, 15.
*(Euvres de Gerbert (ed. A. Olleris; Paris, 1867), Figs. 1-100, following p. 475.
4 R. C. Archibald, Euclid's Book on Divisions of Figures (Cambridge, 1915),
p. 12.
5 Evclides Megarensis .... sex libri priores .... authorc loanne Schevbelio
(Basel, [1550]).
6 P. Rami Scholarvm mathematicorvm libri vnus et triginta (Basel, 1569).
GEOMETRY 421
In the designation of a group of points of equal rank or of the same
roperty in a figure, resort was sometimes taken to the repetition of
le and the same letter, as in the works of Gregory St. Vincent,1
laise Pascal,2 John Wallis,3 and Johann Bernoulli.4
377. The next advancement was the introduction of indices at-
iched to letters, which proved to be an important aid. An apparently
nconscious use of indices is found in Simon Stevin,5 who occasionally
ses dotted letters B, B to indicate points of equal significance ob-
tined in the construction of triangles. In a German translation6 of
bevin made in 1628, the dots are placed beneath the letter B, B.
imilarly, Fr. van Schooten7 in 1649 uses designations for points:
C, 2C, 3C; S, 25, 3S; T, 2T, 3T7; 7, 27, 37 .
his procedure is followed by Leibniz in a letter to Oldenburg8 of
ugust 27, 1676, in which he marks points in a geometric figure by
2, 2B, zB, iD, 2/), sD. The numerals are here much smaller than
ic letters, but are placed on the same level with the letters (see also
549). This same notation is used by Leibniz in other essays9 and
^ain in a treatise of 1677 where he lets a figure move so that in its
3w position the points are marked with double indices like 1© and
'15. In 1679 he introduced a slight innovation by marking the
Dints of the principal curve 36, 6Z>, 96 . . . . , generally yb, the curves
: the entire curve yb. The point 3b when moved yields the points
36, 2 36, 3 36; the surface generated by $6 is marked zyb. Leibniz
sed indices also in his determinant notations (Vol. II, § 547).
1 Gregory St. Vincent, Opus geometricum (Antwerp, 1647), p. 27, etc. See also
arl Bopp, "Die Kegelschnitte des Gregorius a St. Vinccntio" in Abhandlungen
>,r Gesch. d. math. Wissensch., Vol. XX (1907), p. 131, 132, etc.
2 Blaise Pascal, "Lettre de Dettonville a Carcavi," (Euvres completes, Vol. Ill
>aris, 1866), p. 364-85; (Euvres (ed. Faugere; Paris, 1882), Vol. Ill, p. 270-
t6.
3 John Wallis, Operum mathematicorum pars altera (Oxford, 1656), p. 16-160.
4 Johann Bernoulli, Ada eruditorum (1697), Table IV; Opera omnia (1742),
ol. I, p. 192.
8S. Stevin, (Euvres (e"d. A. Girard; Leyden, 1634), Part II, "Cosmographie,"
15.
'See J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 46.
7 F. van Schooten, Geometria a Renato des Cartes (1649), p. 112.
8J. Collins, Commerdum epistolicum (ed. J. B. Biot and F. Lefort, 1856),
113.
9 Leibniz Mathematische Schriften, Vol. V (1858), p. 99-113. See D. Mahnke
BiblMeca mathematica (3d ser.), Vol. XIII (1912-13), p. 250.
422 A HISTORY OF MATHEMATICAL NOTATIONS
I. Newton used dots and strokes for marking fluxions and fluents
(§§ 567, 622). As will be seen, indices of various types occur re-
peatedly in specialized notations of later date. For example, L. Euler1
used in 1748
x1 x" x'"
y' y" y'"
as co-ordinates of points of equal significance. Cotes2 used such strokes
in marking successive arithmetical differences. Monge3 employed
strokes, K\ K", K"\ and also 'K' , "Kf.
378. The introduction of different kinds of type received in-
creased attention in the nineteenth century. Wolfgang Bolyai4 used
Latin and Greek letters to signify quantities, and German letters to
signify points and lines. Thus, ab signifies a line ab infinite on both
sides; ab a line starting at the point a and infinite on the side b; ab a
line starting at b and infinite on the side a; P a plane P extending to
infinity in all directions.
379. A remarkable symbolism, made up of capital letters, lines,
and dots, was devised by L. N. M. Carnot.5 With him,
A, B, C, . . . . marked points
AB, AB marked the segment AB and the circular arc AB
BCD marked that the points B, C, D are collinear, C being
placed between B and D
AB " CD is the point of intersection of the indefinite lines A Bf CD
A BCD marked four points on a circular arc, in the order indi-
cated
AB'CD is the point of intersection of the two arcs A B and CD
F AB* CD is the straight line which passes through the points F
1 L. Euler in Histoire de I* Academic r. d. sciences et d. belles letlres, annee 1748
(Berlin, 1750), p. 175.
2 Roger Cotes, Harmonia mensurarum (Cambridge, 1722), "Aestimatio er-
rorum," p. 25.
3 G. Monge, Miscellanea Taurinensia (1770/73). See H. Wieleitner, Geschichte
der Mathematik, II. Teil, II. Halfte (1921), p. 51.
4 Wolfgangi Bolyai de Bolya, Tentamen (2d ed.), Tom. I (Budapestini, 1897),
p. xi.
6L. N. M. Carnot, De la Correlation des figures de geometric (Paris, an IX •«
1801), p. 40-43.
TB CD
GEOMETRY 423
signifies equipollence, or identity of two objects
marks the angle formed by the straight lines, AB, BC,
B being the vertex
is the angle formed by the two lines AB and CD
&ABC the triangle having the vertices A, B, C
A ABC is a right triangle
ABC is the area of the triangle ABC
\. criticism passed upon Carnot's notation is that it loses its clearness
n complicated constructions.
Reye1 in 1866 proposed the plan of using capital letters, A, B, C,
'or points; the small letters a, 6, c, . . . . , for lines; a, /3, 7, . . . . , for
olanes. This notation has been adopted by Favaro and others.2 Be-
sides, Favaro adopts the signs suggested by H. G. Grassmann,3 AB
'or a straight line terminating in the points A and 5, A a the plane
massing through A and a, aa the point common to a and a; ABC the
olane passing through the points A, By C; afiy the point common to
:he planes a, ft 7, and so on. This notation is adopted also by Cre-
nona,4 and some other writers.
The National Committee on Mathematical Requirements (1923)
-ecornmends (Report, p. 78) the following practice in the lettering of ge-
ometric figures: "Capitals represent the vertices, corresponding small
etters represent opposite sides, corresponding small Greek letters rep-
resent angles, and the primed letters represent the corresponding parts
}f a congruent or similar triangle. This permits speaking of a (alpha)
instead of 'angle AJ and of 'small a1 instead of BC."
380. Sign for spherical excess. — John Caswell writes the spherical
excess c = A+B+C-lSO° thus: "E= Z Z Z -2 J." Letting TT stand
'or the periphery of a great circle, G for the surface of the sphere, R
'or the radius of the sphere, he writes the area A of a spherical tri-
ingle thus:5
1 Reye, Geometric der Lage (Hannover, 1866), p. 7.
2 Antonio Favaro, Lemons de Statique graphiqiie (trad, par Paul Terrier),
L Partie (Paris, 1879), p. 2.
3 II. Grassmann, Ausdehnungslehre (Leipzig, Berlin, 1862).
4 Luigi Cremona, Projective Geometry (trans. Charles Leudesdorf ; Oxford,
L885), chap. i.
6 John Wallis, Treatise of Algebra (London, 1685), Appendix on "Trigonome-
try" by John Caswell, p. 15.
424 A HISTORY OF MATHEMATICAL NOTATIONS
The letter E for spherical excess has retained its place in some books1
to the present time. Legendre,2 in his filaments de geometric (1794,
and in later editions), represents the spherical excess by the letter S.
In a German translation of this work, Crelle8 used for this excess the
sign e. Chauvenet4 used the letter K in his Trigonometry.
381. Symbols in the statement of theorems. — The use of symbols in
the statement of geometric theorems is seldom found in print, but is
sometimes resorted to hi handwriting and in school exercises. It
occurs, however, in William Jones's Synopsis palmariorum, a book
which compresses much in very small space. There one finds, for
instance, "An Z in a Segment >, =, < Semicircle is Acute, Right,
Obtuse."6
To Julius Worpitzky (1835-95), professor at the Friedrich Werder
Gymnasium in Berlin, is due the symbolism S.S.S. to recall that two
triangles are congruent if their three sides are equal, respectively;
and the abbreviations S.W.S., W.S.W. for the other congruence
theorems.6 Occasionally such abbreviations have been used in Amer-
ica, the letter a ("angle") taking the place of the letter W (Winkel),
so that asa and sas are the abbreviations sometimes used. The Na-
tional Committee on Mathematical Requirements, in its Report of
1923, page 79, discourages the use of these abbreviations.
382. Signs for incommensurables. — We have seen (§§ 183, 184)
that Oughtred had a full set of ideographs for the symbolic representa-
tion of Euclid's tenth book on incommensurables. A different set of
signs was employed by J. F. Lorenz7 in his edition of Euclid's Ele-
ments; he used the Latin letter C turned over, as in A O B, to indicate
that A and B are commensurable; while A\JB signified that A and B
are incommensurable; ACL.B signified that the lines A and B are com-
mensurable only in power, i.e., A2 and B2 are commensurable, while
A and B were not; A LT#, that the lines are incommensurable even in
power, i.e., A and B are incommensurable, so are A2 and B2.
1 W. Chauvenet, Elementary Geometry (Philadelphia, 1872), p. 264; A. W.
Phillips and I. Fisher, Elements of Geometry (New York, [1896]), p. 404.
2 A. M. Legendre, Elements de geometric (Paris, 1794), p. 319, n. xi.
8 A. L. Crelle's translation of Legendre's G&mttrie (Berlin, 1822; 2d ed., 1833).
Taken from J. Tropfke, op. cit., Vol. V (1923), p. 160.
4 William Chauvenet, Treatise on Plane and Spherical Trigonometry (Phila-
delphia, 1884), p. 229.
6 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 231.
6 J. Tropfke, op. cit., Vol. IV (2d ed., 1923), p. 18.
7 Johann Friederich Lorenz, Euklid's Elemente (ed. C. B. Mollweide; Halle,
1824), p. xxxii, 194.
GEOMETRY 425
383. Unusual ideographs in elementary geometry. — For "is meas-
ured by" there is found in Hart and Feldman's Geometry1 and in that
rf Auerbach and Walsh2 the sign oc_, in Shutt's Geometry3 the sign IE.
Veronese4 employs ==]== to mark "not equal" line segments.
A horizontal line drawn underneath an equation is used by
Kambly5 to indicate folglich or "therefore"; thus:
384. Algebraic symbols in elementary geometry. — The use of alge-
braic symbols in the solution of geometric problems began at the very
time when the symbols themselves were introduced. In fact, it was
very largely geometrical problems which for their solution created a
iced of algebraic symbols. The use of algebraic symbolism in applied
geometry is seen in the writings of Pacioli, Tartaglia, Cardan, Bom-
belli, Widman, Rudolff, Stifel, Stevin, Vieta, and writers since the
sixteenth century.
It is noteworthy that printed works which contained pictographs
lad also algebraic symbols, but the converse was not always true.
Thus, Barrow's Euclid contained algebraic symbols in superabun-
lance, but no pictographs.
The case was different in works containing a systematic develop-
ment of geometric theory. The geometric works of Euclid, Archi-
nedes, and Apollonius of Perga did not employ algebraic symbolism;
they were purely rhetorical in the form of exposition. Not until the
seventeenth century, in the writings of H6rigone in France, and Ought-
red, Wallis, and Barrow in England, was there a formal translation
rf the geometric classics of antiquity into the language of syncopated
3r symbolic algebra. There were those who deplored this procedure;
tve proceed to outline the struggle between symbolists and rheto-
ricians.
1C. A. Hart and Daniel D. Feldman, Plane Geometry (New York, [1911]),
:>. viii.
2 M. Auerbach and C. B. Walsh, Plane Geometry (Philadelphia, [1920]), p. xi.
3 George C. Shutt, Plane and Solid Geometry [1912], p. 13.
4 Giuseppe Veronese, Elementi di geometria, Part I (3d ed., Verona), p. 12.
BLudwig Kambly, Die Elementar-Mathematik, 2. Thcil: Planimelrie (Breslau,
1876), p. 8, 1. Theil: Arithmelik und Algebra, 38. Aufl. (Breslau, 1906), p. 7.
426 A HISTORY OF MATHEMATICAL NOTATIONS
PAST STRUGGLES BETWEEN SYMBOLISTS AND RHETORI-
CIANS IN ELEMENTARY GEOMETRY
385. For many centuries there has been a conflict between indi-
vidual judgments, on the use of mathematical symbols. On the one
side are those who, in geometry for instance, would employ hardly
any mathematical symbols; on the other side are those who insist on
the use of ideographs and pictographs almost to the exclusion of
ordinary writing. The real merits or defects of the two extreme views
cannot be ascertained by a priori argument; they rest upon experience
and must therefore be sought in the study of the history of our sci-
ence.
The first printed edition of Euclid's Elements and the earliest
translations of Arabic algebras into Latin contained little or no mathe-
matical symbolism.1 During the Renaissance the need of symbolism
disclosed itself more strongly in algebra than in geometry. During the
sixteenth century European algebra developed symbolisms for the
writing of equations, but the arguments and explanations of the
various steps in a solution were written in the ordinary form of verbal
expression.
The seventeenth century witnessed new departures; the symbolic
language of mathematics displaced verbal writing to a much greater
extent than formerly. The movement is exhibited in the writings of
three men: Pierre H6rigone2 in France, William Oughtred3 in Eng-
land, and J. H. Rahn4 in Switzerland. Herigone used in his Cursus
mathematicus of 1634 a large array of new symbols of his own design.
He says in his Preface: "I have invented a new method of making
demonstrations, brief and intelligible, without the use of any lan-
1 Erhard Ratdolt's print of Campanus' Euclid (Venice, 1482). Al-Khowariz-
ml's algebra was translated into Latin by Gerard of Cremona in the twelfth cen-
tury. It was probably this translation that was printed in Libri's Histoire des sci-
ences mathematique en Italie, Vol. I (Paris, 1838), p. 253-97. Another translation
into Latin, made by Robert of Chester, was edited by L. C. Karpinski (New York,
1915). Regarding Latin translations of Al-Khowarizrnf, see also G. Encstrom,
BiUiotheca mathematica (3d scr.), Vol. V (1904), p. 404; A. A. Bjornbo, ibid. (3d
ser.), Vol. VII (1905), p. 239-48; Karpinski, BiUiotheca mathematica (3d ser.),
Vol. XI, p. 125.
2 Pierre Herigone, op. cit., Vol. I-VI (Paris, 1634; 2d ed., 1644).
3 William Oughtred, Claris mathemalicae (London, 1631, and later editions);
also Oughtred's Circles of Proportion (1632), Trigonometric (1657), and minor
works.
4 J. H. Rahn, Teutsche Algebra (Zurich, 1659), Thomas Brancker, An Intro-
duction to Algebra (trans, out of the High-Dutch; London, 1668).
SYMBOLISTS AND RHETORICIANS 427
guage." In England, William Oughtred used over one hundred and
fifty mathematical symbols, many of his own invention. In geometry
Oughtred showed an even greater tendency to introduce extensive
symbolisms than did H6rigone. Oughtred translated the tenth book
of Euclid's Elements into language largely ideographic, using for the
purpose about forty new symbols.1 Some of his readers complained of
the excessive brevity and compactness of the exposition, but Oughtred
never relented. He found in John Wallis an enthusiastic disciple. At
the time of Wallis, representatives of the two schools of mathematical
exposition came into open conflict. In treating the "Conic Sections"2
no one before Wallis had employed such an amount of symbolism.
The philosopher Thomas Hobbes protests emphatically: "And for
.... your Conic Sections, it is so covered over with the scab of sym-
bols, that I had not the patience to examine whether it be well or ill
demonstrated."3 Again Hobbes says: "Symbols are poor unhand-
some, though necessary scaffolds of demonstration";4 he explains
further: "Symbols, though they shorten the writing, yet they do not
make the reader understand it sooner than if it were written in words.
For the conception of the lines and figures .... must proceed from
words either spoken or thought upon. So that there is a double labour
of the mind, one to reduce your symbols to words, which are also
symbols, another to attend to the ideas which they signify. Besides,
if you but consider how none of the ancients ever used any of them in
their published demonstrations of geometry, nor in their books of
arithmetic .... you will not, I think, for the future be so much in
love with them "5 Whether there is really a double translation,
such as Hobbes claims, and also a double labor of interpretation, is a
matter to be determined by experience.
386. Meanwhile the Algebra of Rahn appeared in 1659 in Zurich
and was translated by Brancker into English and published with addi-
tions by John Pell, at London, in 1668. The work contained some
new symbols and also Pell's division of the page into three columns.
He marked the successive steps in the solution so that all steps in the
process are made evident through the aid of symbols, hardly a word
1 Printed in Oughtred's Claims mathematicae (3d ed., 1648, and in the editions
of 1652, 1667, 1603). See our §§ 183, 184, 185.
* John Wallis, Operum mathemalicorum, Pars altera (Oxford), De sectionibus
conicis (1655).
3 Sir William Molesworth, The English Works of Thomas Hobbes, Vol. VII
(London, 1845), p. 316.
4 Ibid., p. 248. 5 IUd., p. 329.
428 A HISTORY OF MATHEMATICAL NOTATIONS
of verbal explanation being necessary. In Switzerland the three -
column arrangement of the page did not receive enthusiastic recep-
tion. In Great Britain it was adopted in a few texts: John Ward's
Young Mathematician's Guide, parts of John Wallis' Treatise of Alge-
bra, and John Kirkby's Arithmetical Institutions. But this almost com-
plete repression of verbal explanation did not become widely and
permanently popular. In the great mathematical works of the seven-
teenth century — the GeomMrie of Descartes; the writings of Pascal,
Fermat, Leibniz; the Principia of Sir Isaac Newton — symbolism was
used in moderation. The struggles in elementary geometry were
more intense. The notations of Oughtred also met with a most
friendly reception from Isaac Barrow, the great teacher of Sir Isaac
Newton, who followed Oughtred even more closely than did Wallis.
In 1655, Barrow brought out an edition of Euclid in Latin and in 1660
an English edition. He had in mind two main objects: first, to reduce
the whole of the Elements into a portable volume and, second, to
gratify those readers who prefer "symbolical" to "verbal reasoning."
During the next half-century Barrow's texts were tried out. In 1713,
John Keill of Oxford edited the Elements of Euclid, in the Preface of
which he criticized Barrow, saying: "Barrow's Demonstrations are
so very short, and are involved in so many notes and symbols, that
they are rendered obscure and difficult to one not versed in Geometry.
There, many propositions, which appear conspicuous in reading
Euclid himself, are made knotty, and scarcely intelligible to learners,
by his Algebraical way of demonstration The Elements of all
Sciences ought to be handled after the most simple Method, and not to
be involved in Symbols, Notes, or obscure Principles, taken else-
where." Keill abstains altogether from the use of symbols. His expo-
sition is quite rhetorical.
William Whiston, who was Newton's successor in the Lucasian
professorship at Cambridge, brought out a school Euclid, an edition
of Tacquet's Euclid which contains only a limited amount of symbol-
ism. A more liberal amount of sign language is found in the geometry
of William Emerson.
Robert Simson's edition of Euclid appeared in 1756. It was a care-
fully edited book and attained a wide reputation. Ambitious to pre-
sent Euclid unmodified, he was careful to avoid all mathematical
signs. The sight of this book would have delighted Hobbes. No scab
of symbols here!
That a reaction to Simson's Euclid would follow was easy to see.
In 1795 John Playfair, of Edinburgh, brought out a school edition
SYMBOLISTS AND RHETORICIANS 429
of Euclid which contains a limited number of symbols. It passed
through many editions in Great Britain and America. D. Cresswell,
of Cambridge, England, expressed himself as follows: "In the demon-
strations of the propositions recourse has been made to symbols,
But these symbols are merely the representatives of certain words and
phrases, which may be substituted for them at pleasure, so as to
render the language employed strictly comfonnable to that of ancient
Geometry. The consequent diminution of the bulk of the whole book
is the least advantage which results from this use of symbols. For
the demonstrations themselves are sooner read and more easily com-
prehended by means of these useful abbreviations; which will, in a
short time, become familiar to the reader, if he is not beforehand per-
fectly well acquainted with them."1 About the same time, Wright2
made free use of symbols and declared: "Those who object to the
introduction of Symbols in Geometry are requested to inspect Bar-
row's Euclid, Emerson's Geometry, etc., where they will discover many
more than are here made use of." "The difficulty," says Babbage,3
"which many students experience in understanding the propositions
relating to ratios as delivered in the fifth book of Euclid, arises en-
tirely from this cause [tedious description] and the facility of com-
prehending their algebraic demonstrations forms a striking contrast
with the prolixity of the geometrical proofs."
In 1831 R. Blakelock, of Cambridge, edited Simson's text in the
symbolical form. Oliver Byrne's Euclid in symbols and colored dia-
grams was not taken seriously, but was regarded a curiosity.4 The
Senate House examinations discouraged the use of symbols. Later
De Morgan wrote: "Those who introduce algebraical symbols into
elementary geometry, destroy the peculiar character of the latter to
1 A Supplement to the Elements of Euclid, Second Edition .... by D. Cresswell,
formerly Fellow of Trinity College (Cambridge, 1825), Preface. Cresswell uses
algebraic symbols and pictographs.
2 J. M. F. Wright, Self-Examination in Euclid (Cambridge, 1829), p. x.
3 Charles Babbage, "On the Influence of Signs in Mathematical Reasoning,"
Transactions Cambridge Philos. Society, Vol. II (1827), p. 330.
4 Oliver Byrne, The Elements of Euclid in which coloured diagrams and symbols
are used (London, 1847). J. Tropfke, op. cit., Vol. IV (1923), p. 29, refers to a
German edition of Euclid by Heinrich Hoffmann, Teutscher Euclides (Jena, 1653),
as using color. The device of using color in geometry goes back to Heron (Opera,
Vol. IV [ed. J. L. Heiberg; Leipzig, 1912], p. 20) who says: "And as a surface one
can imagine every shadow and every color, for which reason the Pythagoreans
called surfaces 'colors.' " Martianus Capella (De nuptiis [ed. Kopp, 1836], No.
708) speaks of surfaces as being "ut est color in corpore."
430 A HISTORY OF MATHEMATICAL NOTATIONS
every student who has any mechanical associations connected wi
those symbols; that is, to every student who has previously used the
in ordinary algebra. Geometrical reasons, and arithmetical procej
have each its own office; to mix the two in elementary instruction,
injurious to the proper acquisition of both."1
The same idea is embodied in Todhunter's edition of Euclid whi<
does not contain even a plus or minus sign, nor a symbolism for pr
portion.
The viewpoint of the opposition is expressed by a writer in tl
London Quarterly Journal of 1864: "The amount of relief which h
been obtained by the simple expedient of applying to the elements
geometry algebraic notation can be told only by those who rernemb
to have painfully pored over the old editions of Simson's Euclid. Tl
practical effect of this is to make a complicated train of reasoning
once intelligible to the eye, though the mind could not take it
without effort."
English geometries of the latter part of the nineteenth centui
and of the present time contain a moderate amount of symbol isr
The extremes as represented by Oughtred and Barrow, on the 01
hand, and by Robert Simson, on the other, are avoided. Thus
conflict in England lasting two hundred and fifty years has ended as
draw. It is a stupendous object-lesson to mathematicians on math
matical symbolism. It is the victory of the golden mean.
387. The movements on the Continent were along the same line
but were less spectacular than in England. In France, about a cei
tury after Hcrigone, Clairaut2 used in his geometry no algebraic sigi
and no pictographs. Bezout3 and Legendre4 employed only a rnodcra
amount of algebraic signs. In Germany, Karsten5 and Segner6 mac
only moderate use of symbols in geometry, but Reyher7 and Loren
1 A. de Morgan, Trigonometry and Double Algebra (1849), p. 92 n.
2 A. C. Clairaut, Siemens de geometric (Paris, 1753; 1st ed., 1741).
3 E. B6zout, Cours de Maihematiques, Tom. I (Paris: n. 6d., 1797), Siemens
gevm&rie.
4 A. M. Legendre, Elements de Geometrie (Paris, 1794).
5 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, I. Theil (Greifswal
1767), p. 205-484.
6 1. A. de Segner, Cursus mathematics Pars I: Elementa arithmetics, ge
metriae et calculi geometrici (editio nova; Halle, 1767).
7 Samuel Reyher s .... Euclides (Kiel, 1698).
8 J. F. Lorenz, Euklid's Elemente, auf's neue herausgegeben von C. B. Mo
weide (5th ed., Halle, 1824; 1st ed., 1781; 2d ed., 1798).
SYMBOLISTS AND RHETORICIANS 431
used extensive notations; Lorenz brought out a very compact edition
of all books of Euclid's Elements.
Our data for the eighteenth and nineteenth centuries have been
drawn mainly from the field of elementary mathematics. A glance at
the higher mathematics indicates that the great mathematicians of
the eighteenth century, Euler, Lagrange, Laplace, used symbolism
freely, but expressed much of their reasoning in ordinary language.
In the nineteenth century, one finds in the field of logic all gradations
from no symbolism to nothing but symbolism. The well-known oppo-
sition of Steiner to Plticker touches the question of sign language.
The experience of the past certainly points to conservatism in the
use of symbols in elementary instruction. In our second volume we
indicate more fully that the same conclusion applies to higher fields.
Individual workers who in elementary fields proposed to express
practically everything in ideographic form have been overruled. It is
a question to be settled not by any one individual, but by large groups
or by representatives of large groups. The problem requires a con-
sensus of opinion, the wisdom of many minds. That widsom dis-
closes itself in the history of the science. The judgment of the past
calls for moderation.
The conclusion reached here may be stated in terms of two school-
boy definitions for salt. One definition is, "Salt is what, if you spill a
cupful into the soup, spoils the soup." The other definition is, "Salt
is what spoils your soup when you don't have any in it."
ALPHABETICAL INDEX
(Numbers refer to paragraphs)
Abacus, 39, 75, 119
Abu Kamil, 273: unknown quantity,
339
Ada eruditorum, extracts from, 197
Adam, Charles, 217, 254, 300, 344
Adams, D., 219, 286, 287
Addition, signs for: general survey of,
200-216; Ahmes papyrus, 200; Al-
Qalasadi, 124; Bakhshali MS, 109;
Diophantus, 102; Greek papyri, 200;
Hindus, 106; Leibniz, 198; el in
Regiomontanus, 126
Additive principle in notation for pow-
ers, 116, 124, 295; in Pacioli, 135; in
Gloriosus, 196
Additive principles: in Babylonia, 1;
in Crete, 32; in Egypt, 19, 49; in
Rome, 46, 49; in Mexico, 49; among
Aztecs, 66
Adrain, R., 287
Aepinus, F. V. T., parentheses, 352
Aggregation of terms: general survey
of, 342-56; by use of dots, 348;
Oughtred, 181, 183, 186, 251; Ro-
manus, 320; RudolfT, 148; Stifel, 148,
153; Wallis, 196. By use of comma,
189, 238; communis radix, 325; Ra.
col. in Scheubel, 159; aggregation of
terms, in radical expressions, 199,
319, 332, 334; redundancy of sym-
bols, 335; signs used by Bombelli,
144, 145; Clavius, 161; Leibniz, 198,
354; Macfarlane, 275; Oughtred,
181, 183, 251, 334; Pacioli. See
Parentheses, Vinculum
Agnesi, M. G., 253, 257
Agrippa von Nettesheim, 97
Ahmes papyrus, 23, 260; addition and
subtraction, 200; equality, 260;
general drawings, 357, 376; unknown
quantity, 339; fractions, 22, 23, 271,
274
Akhmim papyrus, 42
Aladern, J., 92
Alahdab, 118
Al-Battani, 82
Albert, Johann, 207
Alexander, Andreas, 325, 326; aggre-
gation, 343, 344
Alexander, John, 245, 253, 254; equality,
264; use of star, 356
Algebraic symbols in geometry, 384
Algebras, Initius, 325
Al-IIassar, 118, 235, 272; continued
fractions, 118
Ali Aben Ragel, 96
Al-Kalsadi. See Al-Qalasadf
Al-Karkhf, survey of his signs, 116, 339
Al-Khowarizmf, survey of his signs,
115; 271, 290, 385
Allaize, 249
Alligation, symbols for solving prob-
lems in, 133
Al-Madjrltl, 81
Alnasavi, 271
Alphabetic numerals, 28, 29, 30, 36, 38,
45, 46, 87; for fractions, 58, 59; in
India, 76; in Rome, 60, 61
Al-Qalasadf: survey of his signs, 124;
118, 200, 250; equality, 124, 260;
unknown, 339
Alstcd, J. H., 221, 225, 229, 305
Amicable numbers, 218, 230
Anatolius, 117
Anderegg, F., and E. D. Rowe: equiva-
lence, 375
Andre*, D., 95, 243, 285
Andrea, J. V., 263
Angle: general survey of, 360-63; sign
for, in H6rigone, 189, 359; oblique
angle, 363 ; right angle, 363 ; spherical
angle, 363; solid angle, 363; equal
angles, 363
Anianus, 127
Apian, P., 148, 222, 223, 224, 278
Apolionius of Perga, 384
Arabic numerals. See Hindu-Arabic
numerals
Arabs, early, 45; Al-Khow&rizmf, 81,
115, 271, 290, 385; Al-Qalasadf, 118,
124, 200, 250, 260, 339; Al-Madjrltl,
81; Alnasavi, 271; Al-Karkhf, 116,
339; Ali Aben Ragel, 96
433
434
A HISTORY OF MATHEMATICAL NOTATIONS
Arc of circle, 370
Archibald, R. C., 218, 270, 376
Archimedes, 41, 384
Ardliser, Johann, 208, 283; aggregation,
348; radical sign, 328, 332
Arithmetical progression, 248; arith-
metical proportion, 249, 255
Arnauld, Antoine, 249; equality, 266
Aryabha^a, 76
Astronomical signs, relative position of
planets, 358
Athelard of Bath, 81, 82
Attic signs, 33, 34, 35, 84
Auerbach, M., and C. B. Walsh: angle,
361; congruent in geometry, 372;
is measured by, 383
Aurel, Marco, 165, 204, 327
Ayres, John, 225
Aztecs, 66
Babbage, Charles, 386; quoted, 386
Babylonians, 1-15; ideogram for multi-
plication, 217; ideogram for division,
235
Bachet, C. G., 101, 339
Bagza, L., 222
Bailey, M. A., 242
Baillet, J., 41
Bails, Benito, 249
Baker, Alfred, congruence in geometry,
374
Baker, Th.: parallel, 368; perpendicu-
lar, 364; use of star, 356
Bakhshali MS: survey -of signs, 109;
106, 200, 217, 235, 250, 260; equality,
260, 109; unknown quantity, 339
Balam, R., 186, 209, 245, 246, 283, 303;
arithmetical proportion, 249
Balbontin, J. M., 239, 258
Ball, W. W. R., 96
Ballantine, J. P., 90
Bamberg arithmetic, 272
Barlaam, 40
Bar-le-Duc, I. Errard de. See Deidier,
Dounot
Barlow, Peter, 225, 286
Barreme, N., 91
Barrow, Isaac: survey of his signs, 192;
216, 237, 371, 384, 386; aggregation
345, 354; equality, 266; geometrical
proportion, 251, 252; powers, 307
Bartholinus, E., 217
Bartjens, William, 52, 208
Barton, G. A., 9, 10, 12
Beaugrand, Jean de, 302
Beeckman, I., 340
Beguelin, Nic. de, 255
Belidor, Bernard Forest de, 206, 248,
255, 257
Bellavitis, G., 268
Beman, W. W., 244. See also Beman
and Smith
Beman, W. W., and D. E., Smith:
angle, 360; right angle, 363; parallelo-
gram, 365; similar, 372
Benedetti, J. B., 219
Benson, L. S., equivalence, 375
Berlet, Bruno, 148, 326
Bernhard, Max, angle, 362
Bernoulli, Daniel (b. 1700), paren-
theses, 351, 352
Bernoulli, Jakob I (James), 210, 255;
aggregation, 348, 354; equality, 264,
267; radical signs, 334; D as oper-
ator, 366
Bernoulli, Johann I (John), 233, 255,
258, 309, 310, 341; aggregation, 344;
lettering figures, 376; "mem," 365;
use of star, 356; radical expressions,
308; D as operator, 366
Bernoulli, Johann II (b. 1710), 258;
parentheses, 351, 352
Bernoulli, Johann III (b. 1744), 208,
365
Bernoulli, Nicolaus (b. 1687), expo-
nents, 313
Bertrand, Louis, 286
Bettini, Mario, 96, 226, 229
Betz, W., and H. E. Webb, congruent
in geometry, 374
Beutel, Tobias, 208, 273, 283, 292
Beyer, J. H., 277, 283
Bezold, C, 358
Be*zout, E., 91, 241, 248; geometry, 387
Bhaskara: survey of his signs, 110-14;
109, 217, 295; unknown quantity,
339
Bianchini, G., 126, 138, 208, 318
Biart, L., 66
Biernatzki, 71
Billingsley's Euclid, 169, 251
Billy, J. de, 227, 249, 253, 254, 292;
aggregation, 351; exponents, 307;
use of g, 320
Binet, J., angle, 361
Biot, 71
ALPHABETICAL INDEX
435
Birks, John, omicron-sigrna, 307
Bjornbo, A. A., 385
Blakelock, II., 386
Blassiere, J. J., 248, 249; eirele, 367;
parentheses, 351; similar, 372
Blundeville, Th., 91, 223
Bobynin, V. V., 22
Bocthius, 59; apices, 81; proportions,
249, 250
Boeza, L., 273
Boissiere, Claude do, 229
Bolognetti, Pompeo, 145
Bolyai, John, 308
Bolyai, Wolfgang, 212, 268; angle, 361,
362; circle, 367; congruent in geome-
try, 374; different kinds of typo, 378;
equivalence, 376
Bombelli, Itafaelo: survey of his signs,
144, 145; 162, 164, 190,' 384; aggrega-
tion, 344; use of /J, 319, 199
Boinie, 258
Boncompagni, B., 91, 129, 131, 132,
219, 271, 273, 359
Boon, C. F., 270
Borel, E., angle, 361
Borgi (or Borghi) Pietro, survey of his
signs, 133; 223, 278
Bortolotti, E., 47, 138, 145, 344, 351
Bosch, Klaas, 52, 208
Bosnians, H., 160, 162, 172, 176, 297
Boudrot, 249
Bouguer, P., 258
Bourke, J. G., 65
Braces, 353
Brackets, 347; in Bombelli, 351, 352
Brahmagupta : survey of his signs, 106-
8; 76, SO, 112, 114; unknown quanti-
ties, 339
Brancker, Thomas, 194, 237, 252, 307,
386; radical sign, 328, 333, unknown
quantity, 341
Brand is, 88
Brasch, F. E., 125
Brasser, J. It., 343
Briggs, II., 261; decimal fractions, 283;
use of / for root, 322
Brito Rebello, J. I., 56
Bronkhorst, J. (Noviomagus), 97
Brouncker, W., 264
Brown, Hiehard, 35
Briickner, Mac, angle, 362
Brugsch, H., 16, 18, 200
Bryan, G. H., 334, 275
Bubnov, N., 75
Budden, E., similar, 374
Biihler, G., 80
Biirgi, Joost, 278, 283; powers, 296
Bur j a, Abel, radical sign, 331
Bush, W. N., and John B. Clarke:
angle, 363; congruent in geometry,
372; =, 374
Buteon, Jean: survey of his signs, 173;
132, 204, 263; equality, 263
Byerly, \V. E.: angle, 360; parallelo-
gram, 365; right angle 363
Byrne, ().: angle, 362; edition of Eu-
clid, 386; right angle, 363
Cajori', F., 75, 92
Calculus, differential and integral, 365,
377
Caldcrfai (Span, sign), 92
Callet, Fr., 95; similar, 374
Cambuston, H., 275
Campanus' Euclid, 385
Camus, C. E. L., 372
Cantor, Moritz, 27, 28, 31-34, 36, 38,
46, 47, 69, 71, 74, 76, 81, 91, 96, 97,
100, 116, 118, 136, 144, 201, 238, 263,
264, 271, 304, 324, 339; per mille,
274; lettering of figures, 376
Capella, Martianus, 322, 386
Cappelli, A., 48, 51, 93, 94, 208, 274
Caramuel, J., 91, 92; decimal separa-
trix, 262, 283; equality, 205; powers,
303, 305, 300; radical signs, 328;
unknowns, 341
Cardano (Cardan), Ilicroniino: sur-
vey of his signs, 140, 141; 152, 101,
176, 106, 177, 381; aggregation,
343, 351; equality, 140, 260; use of
/£ , 199. 319; use of round parentheses
once, 351
Carlos le-Maur, 96
Carmichael, Robert, calculus, 348
Carnot. L. X. M.: angle, 705; geo-
metric notation, 379
Carra do Vaux, 75
Carre, L., 255, 266; parentheses, 351
Cartan, E., 247
Casati, P., 273
Casey, John: angle, 360; congruent in
geometry, 374
Cassany, F., 254
Castillon, G. F., 286; radical signs, 330
436
A HISTORY OF MATHEMATICAL NOTATIONS
Casting out the 9's, 7's, ll's, 218, 225
Castle, F., variation, 259
Caswcll, John: circle, 367; equiangu-
lar, 360; parallel, 368; perimeter,
367; perpendicular, 364; spherical
excess, 380
Cataldi, P. A., 148, 170, 196, 296;
D as an operator, 366; unknown
quantities, 339
Catelan, Abb6, 266
Cauchy, A. L. : negative numerals, 90;
principal values of a*, 312
Cavalieri, B. : survey of his signs, 179;
204, 261; aggregation, 344
Cayley, A., 275
Census, word for z2, 116, 134
Chace, A. B., 23
Chalosse, 219, 221
Chapelle, De la, 223, 255; radical sign,
331
Chauvenet, W., 259; algebraic symbols,
371; right angle, 363; spherical ex-
cess, 380
Chelucci, Paolino, 249, 285
China, 69, 119, 120; unknown quantity,
339
Chrystal, G., variation, 259
Chu Shih-Chieh, survey of his signs,
119, 120
Chuquet, N. : survey of his signs, 129-
32; 117, 145, 164, 190, 200, 219,
220, 222, 230, 296, 308; aggregation,
344, 350; use of square, 132, 357;
use of #, 199 318
Churchill, Randolph, 286
Cifrdo (Portuguese sign), 94
Circle: arcs of, 359; pictograph for,
357, 359, 367, 371; as a numeral, 21
Ciscar, G., 286
Clairaut, A. C., 258; his geometry, 387;
parentheses, 352
Clark, Gilbert, 186, 248
Clarke, H., 286, 289
Clavius, C.: survey of his signs, 161;
91, 179, 204, 205, 219, 222, 250, 279,
300; aggregation, 351; decimal point,
280; plus and minus, 199; powers,
300; radical sign, 327
Cobb, Sam, 253
Cocker, Edward: aggregation, 347, 348
Codex Vigilanus, 80,
Coefficients: letters as coefficients,
176-78, written after the literal part,
179; written above the line, 307
Cole, John, 259
Colebrooke, H. Th., 76, 80, 91, 106,
107, 108, 110, 112-14
Collins, John, 195, 199, 237, 196, 252,
305, 307, 308, 344; aggregation, 344,
350
Colomera y Rodrfguez, 92
Colon: for aggregation, 332; separa-
trix, 245; sign for ratio, 244, 251,
258, 259
Color: used in marking unknowns, 107,
108, 112, 114; colored diagrams, 386;
colored quipu, 62, 64
Colson, John, 253; negative numerals,
90.
Comma: for aggregation, 334, 342, 349;
for multiplication, 232, 233; for ratio,
256, 257; decimal fractions, 278, 282,
283, 284, 285 .
Condorcet, N. C. de, parentheses, 352,
354
Congruent, signs in geometry, 372-75
Continued fractions, 118, 273; in John
of Meurs, 271; in Wallis, 196
Copernicus, N., 157
Corachan, J. B., 207, 250
Cortaziir, J., 286
Cosa, 290, 318; in Buteon, 173; in
Chuquet, 131; in De la Roche, 132;
in Pacioli, 134, 136, 339; in Rudolff,
149
Cossali, P., 249; aggregation, 345, 354,
355
Cotes, Roger, 307
Courtivron, le Marquis de, parentheses,
352
Craig, John, 252, 253, 301; aggrega-
tion, 345, 347
Cramer, G., aggregation, 345, 354, 355
Crelle, A. L., spherical excess, 380
Cremona, L., coincides with, 374
Cresswell, D., 386
Creszfeldt, M. C., 91
Cretan numerals, 32
Crocker, E., 248
Cruquius, N., 208
Crusius, D. A., 96, 208
Crusoe, G. E., 222, 226
Cube of a number, Babylonians, 15
Cuentos(l 'millions"), abbreviation for, 92
Cuneiform symbols, 1-15
Cunha, J. A. da. See Da Cunha, J. A.
Cunn, Samuel, aggregation, 210/345
ALPHABETICAL INDEX
437
Curtze, M., 81, 85, 91, 123, 126, 138,
219, 250, 290, 318, 325, 340, 359
Gushing, F. H., 65
Dacia, Petrus de, 91
Da Cunha, J. A., 210, 236, 307
Dagomari, P., 91
D'Alembert, J., 258; angle, 360, 363;
imaginary j/— 1, 346; parentheses,
352; similar, 372
Dash. See Line
Dasypodius, C., 53
Datta, B., 75
Da vies, Charles, 287; equivalence, 375
Davies, Charles, and W. G. Peck:
equivalence, 375, repeating deci-
mals, 289
Davila, M., 92
Debeaune, F., 264, 301
De Bessy, Frenicle, 266
Dechales, G. F. M., 206, 225; decimals,
283; equality, 266; powers, 201
Decimal fractions: survey of, 276-89;
186, 351; in Leibniz, 537; in Stevin,
162; in Wallis, 196; repeating deci-
mals, 289
Decimal scale: Babylonian, 3; Egyp-
tian, 16; in general, 58; North Ameri-
can Indians, 67
Decimal separatrix: colon, 245; com-
ma, 282, 284, 286; point, 287, 288;
point in Austria, 288
Dee, John: survey of his signs, 169;
205, 251, 254; radical sign, 327
De Graaf, A. See Graaf, Abraham de
Degrees, minutes, and seconds, 55; in
Regiomontanus, 126, 127
De Gua. See Gua, De
Deidier, L'Abbe, 249, 257, 269, 285,
300, 351
De Lagny. See Lagny, T. F. de
Delahire, 254, 258, 264
De la Loubere, 331
Delambre, 87
De la Roche, E.: survey of his signs,
132; 319, radical notation, 199; use
of square, 132, 357
Del Sodo, Giovanni, 139
De Moivre, A., 206, 207, 257; aggrega-
tion, 354
De Montigny. See Montigny, De
De Morgan. Augustus, 202, 276, 278,
283; algebraic symbols in geometry,
386: complicated exponents, 313;
decimate, 286, 287; equality, 268;
radical signs, 331; solidus, 275
Demotic numerals, 16, 18
Descartes, Rend: survey of his signs,
191; 177, 192, 196, 205, 207, 209, 210,
217, 256, 386; aggregation, 344, 353;
equality, 264, 265, 300; exponential
notation, 294, 298-300, 302-4, 315;
geometrical proportion, 254; plus
or minus, 262; radical sign, 329, 332,
333; unknown quantities, 339, 340;
use of a star, 356
Despiau, L., 248
Determinants, suffix notation in Leib-
niz, 198
De Witt, James, 210, 264
Dibuadius, Christophorus, 273, 327,
332; aggregation, 348
Dickson, W., 286
Diderot, Denys, 255; Encydoptdie, 254
Didier. See Bar-ie-Duc
Diez de la Calle, Juan, 92
Diez freyle, Juan, 290
Difference (arithmetical): •"" — symbol
for, 164, 177, 262; in Leibniz, 198,
344; in Oughtred, 184, 372
Digges, Leonard and Thomas: survey
of their signs, 170; 205, 221, 339;
aggregation, 343; equality, 263;
powers, 296; radical signs, 199, 334
Dilworth, Th., 91, 246, 287
Diophantus: survey of his signs, 101-5;
41, 87, 111, 117, 121, 124, 135, 200,
201, 217, 235; equality, 260, 104, 263;
fractions, 274; powers of unknown,
295, 308, 339
Distributive, ideogram of Babylonians,
15
Division, signs for: survey of, 235-47;
Babylonians, 15; Egyptians, 26;
Bakhshall, 109; Diophantus, 104;
Leonardo of Pisa, 235, 122; Leibniz,
197, 198; Oughtred, 186; Wallis, 196;
complex numbers, 247; critical esti-
mate, 243; order of operations in-
volving -T- and X, 242: relative
position of dividend and divisor,
241; scratch method, 196; -*-, 237,
240; :, 238, 240; £> 154, 162, 236
Dixon, R. B., 65
Dodson, James, 354
Doehlmann, Karl, angle, 362
Dot: aggregation, 181, 183, 251, 348;
as radical sign, 324-26; as separatrix
in decimal fractions, 279, 283-85;
438
A HISTORY OF MATHEMATICAL NOTATIONS
demand for, 251; for ratio, 244; geo-
metrical ratii, 251-53; in complex
numbers, 247; multiplication, in
Bhaskara, 112, 217; in later writers,
188, 233, 287, 288; negative number,
107; to represent zero, 109
Dounot (Deidier, or Bar-lc-Duc), 300,
351
Drachm or dragma, 149, 151, 158, 293
Drobisch, M. W., 202
Ducange, 87
Dulaurens, F., 255; angle, 360; equal-
ity, 263; majus, 263; parallel, 368;
perpendicular, 364; pictographs, 365;
solids, 371
Duraesnil, G., 96
Duodecimal scale, 3; among Romans,
58,59
Du Pasquier, L. Gustavo, 269
Dupuis, N. F., 95; congruent in geome-
try, 374; right angle, 363
Durell, Clement V., angle, 361
Durell, Fletcher, 375
Du Se*jour. See Se*jour, Du
Edwards, R. W. K, 258
Eells, W. C., 67
Egypt, 16; multiplication, 217; square-
root sign, 100
Egyptian numerals, 16-23
Einstein, A., 215
Eisenlohr, A., 23
El-Hassar. See al-Hassar
Emerson, W., 248, 249; angle, 360;
geometry of, 386; parallel, 368;
perpendicular, 364; variation, 259
Enestrom, G., 91, 135, 136, 139, 150,
141, 271, 278, 325, 339, 340, 351,
385
Enriques, F., angle, 361
Equal and parallel, 369
Equality: survey of, 260-70; Ahmes
papyrus, 260; al-Qalasadi, 124; Bakh-
shall MS; 109; Buteon, 173; in
Bolognetti, 145; Cardan, 140; dash
in Regiomontanus, 126; dash in
Ghaligai, 139; dash in Pacioli, 138;
Descartes, 191, 300, 363: Digges, 170,
263; Diophantus, 104; Harriot, 188;
H6rigone, 189; in proportion, 251,
256; Recorde, 167
Equivalence, 375
Eratosthenes, 41
Etruian signs, 46, 49
Eucken, A., 270
Euclid's Elements, 158, 166, 169, 179,
216, 318, 384, 385; Newton's anno-
tation, 192; Barrow's editions, 192;
Billingsley's edition, 251; Elements
(Book X), 318, 332; lines for magni-
tudes, 359
Euler, L., 387; aggregation, 350, 352,
354; imaginary exponents, 309; in-
dices in lettering, 377; lettering of
triangle, 194; origin of /, 324;
powers, 304; imaginary j/ — 1, 346
Eutocius, 41
Evans, A. J., 32
Exponents: survey of, 296-315; 129,
131; Bombelli, 144, 162; Chuquet,
131; Descartes, 191; Leibniz, 198;
Nunez, 165; Stevin, 162; general ex-
ponents in Wallis, 195; fractional,
123, 129, 131, 162, 164, 196; nega-
tive, 131, 195, 308, 311; placed before
the base, 198; placed on line in
He*rigone, 189; Roman numerals
placed above the line in Hume, 190
Eygaguirre, S. F., 222
Eysenhut, 203
Factoring, notation for process in
Wallis, 196
Fakhri, 339
Falkenstein, K., 127
False positions, 202, 218, 219
Favaro, A.: angle, 361; use of differ-
ent letters, 379
Faye, P. L., 65
Feliciano, F., 222
Fenn, Joseph, 259; angle, 362; circle,
367; right angle, 363; solids, 371;
use of star, 356
Fermat, P., 101, 206, 261, 386; coef-
ficients, 307; equality, 265; powers,
307
Ferroni, P., aggregation, 354
Fialkowski, N., 288; angle, 362; similar,
372
Fiedler, W.: angle, 360, 362; congruent
in geometry, 372; equal and parallel,
369
Fine, O., 222, 229
Fischer, E. G., 270
"Fisher, George" (Mrs. Slack), 287
Fisher, G. E., and I. J. Schwatt, 213,
242
Fludd, Robert, 204
ALPHABETICAL INDEX
439
Follinus, H., 208, 221, 222; aggrega-
tion, 351; radical signs, 334
Fontaine, A., imaginary i/ — 1, 346
Ford, W. B., negative numerals, 90
Fortunatus, F., 207, 255
Foster, Mark, 254
Foster, S., 186, 251; decimals, 186, 283;
equality, 264; parentheses, 351; pow-
ers, 306
Foucher, L'Abbe*, 249, 255
Fournier, C. F., 248, 249
Fractions: common fractions (survey
of), 271-75; Babylonian, 12, 13, 14,
15; addition and subtraction of, 222;
Bakhshali MS, 109; complex frac-
tions in Stevin, 188; Diophantus,
104; division of, 224; duodecimal,
58, 59, 61; Egyptian, 18, 22, 23, 24;
fractional line, 122, 235, 272, 273,
391; fraction not a ratio, 245; Greeks,
41, 42, 104; Hindus, 106, 113, 235;
juxtaposition means addition, 217;
in Austrian cask measure, 89; in
Recorde, 167; Leibniz, 197, 198;
multiplication, 224; Romans, 58, 59;
special symbol for simple fractions,
274; : to denote fractions, 244; unit
fractions, 22, 41, 42. See Decimal
fractions
Frank, A. von, angle, 363
Frank, Sebastian, 55
Frenicle de Bessy, 266
Fricke, R., use of a star, 356
Friedlein, G., 33, 46, 58, 59, 60, 74, 97,
200
Frisch, Chr., 278, 296
Frischauf, J., similar and congruent,
372
Frisi, P., 259
Fuss, N., 259; angle, 360; parallel, 368
Galileo G., 250, 261, 339
Gallimard, J. E., 236, 239, 255; equal-
ity, 269
Ganesa, 110
Gangad'hara, 91
Garcia, Florentine, 237, 258
Gardiner, W., 285, 367
Garner, J. L., 66
Gauss, K. F., powers, 304, 314
Gausz, F. G., 286
Gebhardt, M., 159
Gelcich, E., 307
Geminus, 41
Gemna Frisius, 91, 219, 222
Geometrical progression, 248; propor-
tion, 249, 250
Geometrical (pictograph) symbols, 189;
in Rich. Rawlinson, 194
Geometry: survey of symbols, 357-87;
symbols in statement of theorems,
381
Gerard, Juan, 249, 258
Gerard of Cremona, 118, 290, 318, 385
Gerbert (Pope Silvester 11), 61, 322,
376
Gerhardt, C. I., 121, 147, 149, 202, 203,
310, 325, 326, 372
Gernardus, 272
Geysius, J., 196, 305, 341
Ghaligai, FT.: survey of his signs, 139;
126, 138, 219, 226; equality, 139,
260; the letter ft, 199, 319
Gherli, O., 285
Ghetaldi, M., 307
Giannini, P., 259
Gibson, Thomas, 233
Ginsburg, J., 79, 280
Girana Tarragones, H., 55
Girard, A.: survey of his signs, 164;
162, 163, 208, 210, 217, 296; aggrega-
tion, 351; coefficients, 307; decimal ,
fractions, 283; difference, 262; pow-
ers, 322; radicals, 329-31, 334
Girault de Koudou (or Keroudou),
Abb6, 255
Glaisherj J. W. L., 202, 208, 275, 282;
complicated exponents, 313
Glorioso, C., 196, 204, 263; unknowns,
339
Gobar numerals, 86, 87, 88, 129
Goldbach, C., 309, 379; figurate num-
bers, 381; exponents, 313
Golius, Jacob, 263
Gonzalez Davila, Gil, 92
Gonzalo de las Casas, J., 92
Gordon, Cosmo, 134
Gosselin, G.: survey of his signs, 174;
204, 320; use of ft, 320; use of capital
L, 174, 175, 290
Gow, J., 26, 41, 359
Graaf, Abraham de, 223, 232. 254;
equality, 263, 264; "mem,'' 365;
radicals, 336; use of star, 356; = for
difference, 262; D as an operator,
366
440
A HISTORY OF MATHEMATICAL NOTATIONS
Grammateus (Heinrich Schreiber) : sur-
vey of his signs, 147; 160, 203, 205,
208, 318
Grandi, Guido, 255
Grassmann, H. G. : congruent in geom-
etry, 374; equal and parallel, 369
Greater or less, Oughtred, 183, 186.
See Inequality
Greek cross, 205
Greek numerals, 33-44, 92; algebra in
Planudes, 121
Green, F. W., 16
Greenhill, A. G., use of star, 356;
approximately equal, 270
Greenstreet, W. J., 331
Greenwood, I., 287
Gregory, David, 252, 257, 304, 308;
aggregation, 345
Gregory, Duncan F., 314
Gregory, James, 207, 252, 256, 268;
radical signs, 334
Gregory St. Vincent, 261
Griev, W., 287
Griffith, F. L., 16, 18
Grisio, M., 196
Grosse, H., 208
Grotefend, 1, 47
Grunert, J. A., 47, 208
Gua, De, 232; aggregation, 350;
use of a star, 356
Guisn6e, 266
Gunther, S., 152, 277
Haan, Bierens de, 91, 164
Haseler, J. F., 368; geometric con-
gruence, 372
Haglund, G., 287
Halcke, P., 208
Hall, H. S., and F. H. Stevens: angle,
360; right angle, 363; parallel, 368
Halley, E., 207, 304; aggregation, 345;
use of star, 356
Halliwell, J. O., 91, 305
Halma, N., 43, 44
Halsted, G. B., 287, 375; angle, 362,
363; arcs, 370; pictographs, 365;
similar and congruent, 372, 374;
symmetry, 371
Hamilton, W. R., arcs, 370
Hammond, Nathaniel, 307
Hankel, H., 33, 59
Harmonic progression, 248
Harrington, M. B., 65
Harriot, Th.: survey of his signs, 188;
156. 192, 196, 205, 217, 233; aggre-
gation, 344, 345: equality, 261, 266,
268; greater or less, 188, 360; repe-
tition of factors, 305; radicals, 329
Harris, John, 253
Harsdorffer, P., 96
Hart, C. A., and D. D. Feldman: arcs,
370; equivalent, 375; is measured by,
383; similar, 372
Hartwell, R., 91
Hatton, Edward, 253, 307; angle, 360,
363; radical signs, 328, 331
Hatton, J. L. S., angle, 362
Hauber,K.F.,307
Hawkes, John, 248
Hawkes, Luby and Teuton, 242
Hawkins, John, 252
Heath, Sir Thomas, 101, 103, 104, 105,
116, 216
Hebrew numerals, 29-31, 36
Heiberg, J. L., 41, 43, 44, 84, 88, 386
Heilbronner, J. C., 40, 47, 97
Heinlin, J. J., 291
Hemelings, J., 208
Henrici, J., and P. Treutlein: angle,
361; equal and parallel, 369; similar,
372
Henrion, D., 178
Henry, C., 176, 204, 263
H£rigone, P.: survey of his signs, 189;
198, 206, 209, 221, 232, 245, 385,
387; angle, 189, 360; arc of circle,
370; circle, 367; equality, 263; great-
er than, 263; perpendicular, 364;
pictographs, 189, 359, 365; powers,
297, 298, 301; radical signs, 189, 334;
ratio, 254; right angle, 363
Hermann, J., 255; parentheses, 351
Herodianic signs, 33, 38
Heron of Alexandria, 41, 103, 201, 271;
circle, 367; parallel, 368; pictograph,
357; colored surfaces, 386
Hieratic numerals, 16, 18, 23, 24, 25, 36,
201
Hieroglyphic numerals, 16, 17, 18, 22;
problem in Ahmes papyrus trans-
lated into hieroglyphic writing, 200
Hill, George A., similar and congruent,
372
Hill, G. F., 74, 80-82, 89
Hilprecht, H. V., 6, 10, 15, 200, 217, 235
Hincks, E., 4, 5
ALPHABETICAL INDEX
441
Hindenburg, C. F.. 207, 208; aggrega-
tion, 354; radical signs, 331
Hindu algebra, 107, 200; division and
fractions, 235
Hindu- Arabic numerals: survey of, 74-
99; 54; al-Qalasadt, 124; Al-Khowa-
rizmt, 115; Chuquet, 129; first oc-
currences, 79, 80; forms, 81-88, 128:
Hindu-Arabic notation, 196; local
value, 78; shape of figure five, 56,
127; shape of zero in Digges, 170;
shape of figure one in Treviso arith-
metic, 86
Hipparchus, 44
Hippocrates of Chios, lettering figures,
376
Hire, De la, 254, 258, 264, 341
Hobbes, Thomas, controversy with
Wallis, 385
Hodgson, James, angle, 360
Hoecke, van der, survey of his signs,
150; 147, 204, 319
Hoernle, A. F. R., 109
Hoffmann, H., 386
Holzmann, W. See Xylander
Hopkins, G. Irving: angle, 360; right
angle, 363; parallelogram, 365; B,
374
Hoppe, E., 351
Horrebowius, P., 232
Horsley, S., 286
Hortega, Juan de, 207, 219, 221, 222,
223, 225
Hospital, L', 206, 255, 266
Hoste, P., 266
Hostus, M., 97
Houel, G. J., 95
Hiibsch, J. G. G., 208, 232
Hudde, J., 264, 307
Huguetan, Gilles, 132
Huips, Frans van der, 266
Hultsch, Fr., 41, 59, 272, 357
Humbert, G., 51
Humboldt, Alex, von, 49, 87, 88
Hume, James: survey of his signs, 190;
206, 297, 298, 302; parentheses, 351;
use of $, 320 •
Hunt, N., 206, 225
Huntington, E. V., 213, 214
Hutton, Ch., 91, 107, 159, 286, 351;
angle, 360
Huygens, Chr., 206, 208, 254, equality,
264; powers, 301, 303, 304, 307, 310
Hypatia, 117
Hypsicles, 44
Ibn Albanna, 118
Ibn Almuncim, 118
Ibn Khaldun, 118
Identity, Riemann's sign, 374
Ideographs, 385; unusual ones in ele-
mentary geometry, . 383
Ifowan as-saft, 83
Illing, C. C., 208
Imaginary y' — 1 or j/( — 1) or i/— 1,
346
Incommensurable: survey of, 382; sign
for, in Oughtred, 183, 184; sign for, in
J. F. Lorenz, 382
Inequality (greater or less) : in Harriot,
188; in H6rigone, 189; in Oughtred,
182. See Greater or less
Infinity, Wallis' sign oo, 196
Isidorus of Seville, 80
Izquierdo, G., 248, 249, 258
Jackson, L. L., 208; quoted, 199
Jacobs, H. von, 96
Jager, R., 283
Japanese numerals, 71, 73
Jeake, S., 219, 223, 245, 249, 254, 284;
radical signs, 328
Jenkinson, H., 74
Jess, Zachariah, 246
John of Seville, 271, 290, 318
Johnson, John, 283
Johnson's Arithmetic, 186, 244
Jones, William, 210, 308; angle, 360,
363; parallel, 368; perpendicular, 364;
pictographs in statement of geomet-
ric theorems, 381; use of a star, 356
Jordan, C., use of s, 374
Juxtaposition, indicating addition, 102;
indicating multiplication, 122, 217
Kastner, A. G., 40
Kalcheim, Wilhelm von, 277
Kambly, L.: angle, 360; arc of circle,
370; horizontal line for "therefore,"
383; parallel, 368; similar and con-
gruent, 372
Karabacek, 80
Karpinski, L. C., 42, 48, 74, 79-81, 87,
92, 115, 116, 122, 159, 208, 266, 271,
273, 287, 385
442
A HISTORY OF MATHEMATICAL NOTATIONS
Karsten, W. J. G.: algebraic symbols,
371; parallel, 368; similar, 372;
signs in geometry, 387; division, 275
Kaye, G. R., 75, 76, 77, 80, 109, 250
Kegel, J. M., 208
Keill, J. : edition of Euclid, 386
Kepler, J., 261, 278, 283; astronomical
signs, 358; powers, 296
Kersey, John, 248, 251, 304, 307;
aggregation, 345; angle, 360; circle,
367; parallel, 368; perpendicular,
364; pictographs, 365; radical signs,
332, 335; right angle, 363
Kinckhuysen, G., 264, 341; use of star,
356
Kirkby, John, 245, 248, 386; arithmet-
ical proportion, 249; sign for evolu-
tion, 328
Kirkman, T. P., 240, 286; aggregation,
354
Klebitius, WiL, 160, 207
Klugel, G. S., 47, 208; angle, 360;
"mem," 365; pictographs, 359, 360;
similar, 372
Knots records; in Peru, 62-64; in China,
69
Knott, C. G., 282
Kobel, J., 55
Konig, J. S., aggregation, 354
Koppe, K., similar and congruent, 372
Kosegarten, 88
Kowalewski, G., 211
Kratzer, A., 270, 271
Krause, 246
Kresa, J., 206, 254
Kritter, J. A., 208
Krogh, G. C., 208
Kronecker, L., 374; [a], 211
Kubitschek, 34
La Caille, Nicolas Louis de, 258
Lagny, T. F., de, 258, 266, 268; radical
signs, 330, 331; use of a star, 356
Lagrange, J., 387; parentheses, 352, 354
Lalande, F. de, 95; aggregation, 354
Lampridius, Aelius, 51
Lamy, B., 206, 248, 249, 255, 257, 264
Landen, John, circular arc, 370
Lansberg, Philip, 250
Laplace, P. S., 99^387 ; aggregation, 354;
imaginary J/ — 1, 346
Latin cross, 205, 206
Lotus ("side"), 290; survey of, 322;
use of I for x, 186, 322; use of L for
powers and roots, 174, 175
Layng, A. E., 246; angle, 360; paral-
lelogram, 365
Lee, Chauncey, 221, 254, 287
Leechman, J. D., 65
Legendre, A. M., 231; aggregation,
354; angle, 363; algebraic signs, 371;
geometry, 387
Leibniz, G. W., 197, 198, 233, 237, 341,
386; aggregation, 344, 349-51, 354;
dot for multiplication, 285; equality,
263, 266, 267; fractions, 275; geo-
metrical proportion, 255, 258, 259;
geometric congruence, 372; lettering
figures, 377; powers, 303, 304;
quotations from, 197, 198, 259; radi-
cal sign, 331; signs for division, 238,
244, 246; variable exponents, 310
Lemoch, I., 288
Lemos, M., 91
Lenormant, F., 5
Leonardo of Pisa: survey of his signs,
122; 91, 134, 219, 220, 235; fractions,
271, 273; letters for numbers, 351;
lettering figures, 376; radix, 290, 292,
318
Lepsius, R., 5
Leslie, John, 371
Less than, 183. See Inequality
Lettering of geometric figures, 376
Letters: use of, for aggregation, 342,
343; capital, as coefficients by Vieta,
176; Cardan, 141; Descartes, 191;
Leibniz, 198; Rudolff, 148; small,
by Harriot, 188; lettering figures,
376
Leudesdorf, Ch., 379
Leupold, J., 96
Le Vavasseur, R., 211
Lcybourn, William, 252, 292; angle,
360
Libri, 91, 116, 385
Lieber, II., and F. von Luhmann:
angle, 360; arc of circle, 370
Lietzmann, W., 96
Li6vano, L, 248, 249, 258
Line: fractional line, 235, 239; as sign
of equality, 126, 138, 139; as sign of
division, 235; as sign of aggregation.
See Vinculum
Lipka, J., equal approximate, 373
Lobachevski, angle of parallelism, 363
ALPHABETICAL INDEX
443
Local value (principle of) : Babylonians,
5, 78; Hebrews, 31; Hindus, 78, 88;
Maya, 68, 78; Neophytos, 88; Turks,
84
Locke, L. L., 62, 63
Long, Edith, and W. C. Brenke, con-
gruent in geometry, 372
Loomis, Elias, 287
Lorenz, J. F., 216; geometry, 387;
incommensurables, 382; similar, 372
Lorey, Adolf: equal and parallel, 369;
similar and congruent, 372
Loubore, De la, 331
Louville, Chevalier de, 258
Lucas, Edouard, 96
Lucas, Lossius, 225, 229
Ludolph van Ceulen, 148, 208, 209,
223; aggregation, 344, 348, 349;
radical sign, 332
Lumholtz, K., 65
Lutz, H. F., 12, 13
Lyman, E. A., parallelogram, 365
Lyte, H., 186
Maandelykse Mathematische Liefheb-
berye, 330, 336
Macdonald, J. W., equivalent, 375
Macdonald, W. R., 282
McDougall, A. H. : congruent in geome-
try, 374; similar, 374
Macfarlane, A., 275
Mach, E., right angle, 368
Maclaurin, Colin, 240, 248; aggrega-
tion, 345; use of a star, 356
McMahon, James, equivalent, 375
Macnie, J., equivalent, 375
Mairan, Jean Jaques d'Orton de, 255
Mako, Paulus, 259, 288
Mai, Arabic for x\ 116, 290
Malcolm, A., 248
Manfredi, Gabriele, 257, 331; paren-
theses, 351
Mangoldt, Hans von, angle, 362
Marini, 49
Marquardt, J., 51
Marre, Aristide, 129
Marsh, John, 289
Martin, T. H., 97
Mason, C., 307
Masterson, Thomas, 171, 278
Mauduit, A. R., 259; perpendicular,
364; similar, 372
Maupertuis, P. L., 255; aggregation.
351, 352
Maurolicus, Fr., 303; pictographs in
geometry, 359; use of g, 319
Maya, 68, 5
Mehler, F. G., angle, 360
Meibomius, M., 251
Meissner, Bruno, 14, 15
Meissner, H., 283
"Mem," Hebrew letter for rectangle,
365, 366
Mengoli, Petro, 206, 254, 301
Menher, 148
Me>ay, Ch., 213
Mercastel, J. B. A. de, on ratio, 254,
256
M creator, N., 252; use of star, 194;
decimals, 283
Mersenne, M., 209, 266, 273, 301, 302,
339
Metius, Adrian, 186, 225; circle, 367;
pictographs, 359, 371
Meurs, John of, 271
Meyer, Friedrich, angle, 362
Meyer, H., 287
Meyers, G. W., congruent in geometry,
372; e, 374
Michelsen, J. A. C., 206
Mikami, Yoshio, 119, 120
Milinowski, A., angle, 362
Milne, W. J., equivalent, 375
Minus sign: survey of, 208-16; in
Bombelli, 144; Buteon, 173; Cava-
lieri, 179; Cardan, 140; Clavius, 161;
Diophantus, 103; Gosselin, 174;
H6rigone, 189; Pacioli, 134; Peletier,
172; Recorde, 167; Regiomontamis,
126, 208; sign m, 131, 132, 134, 142,
172-74, 200; sign ~, 189; sign -?-,
164, 208, 508; Tartaglia, 142, 143;
Vieta, 176; not used in early arith-
metics, 158
Mitchell, J., 286
Mobius, A. F., angle, 361
Mohammed, 45
Molesworth, W., 385
Molk, J., 211
Moller, G., 16, 18, 21
Mollweide, C. B., 47, 216; similar and
congruent, 372
Molyneux, W., 283; radical signs, 334
Mommsen, Th., 46, 51
Monconys, De, 263
444
A HISTORY OF MATHEMATICAL NOTATIONS
Monge, G., lettering, 377
Monich, 246
Monsante, L., 258, 286
Montigny, Ch. de, parentheses, 352
Moore, Jonas, 186, 248; aggregation,
348; arithmetical proportion, 249;
decimals, 286; geometrical propor-
tion, 251; radicals, 332
Moraes Silva, Antonio de, 94
Morley, S. G,, 68
Moxon, J., 231, 236, 303
Moya, P6rez de: 92, 204, 221, 223, 294;
use of #, 320, 321, 326
Mozhnik, F. S., 288
Miiller, A., 96
Muller, C., 41
Muller, G., angle, 360
Muller, O., 49
Multiplication: survey of signs, 217-
34; Bakhshall MS, 109; Cavalieri,
179; comma ia H6rigone, 189; in
Leibniz, 197, 198, 232, 536; cross-
multiplication marked by X, 141,
165; Diophantus, 102' dot, 112, 188,
233, 287, 288; Hindus, 107, 112:
order of operations involving -r and
X, 242; Waliis, 196; of integers, 226,
229; Stevin, 162; Stifel, 154; X, 186,
195, 288; in Oughtred, 180, 186, 231;
in Leibniz, 197, 198; in in Vieta,
176-78, 186; ^ in Leibniz, 198 j star
used by Rahn, 194; juxtaposition,
122, 217
Multiplicative principles, in numeral
system : Aztecs, 66; Babylonians, 1;
Cretans, 32; Egyptians, 19, 21; Ro-
mans, 50, 51, 55; Al-Kharkhf, 116; in
algebraic notation, 101, 111, 116, 135,
142
Musschenbroek, van, 267
Nagl, A., 34, 64, 85, 89
Nallino, C. A., 82
Napier, John, 196, 218, 231, 261, 261;
decimal point, 195, 282; line symbol-
ism for roots, 323, 199
National Committee on Mathematical
Requirements (in U.S.), 243, 288;
angle, 360; radical signs, 338
Nau, F., 79
Negative number, sign for: Bakshall,
109; Hindu, 106
Nemorarius, Jordanus, 272
Neomagus. See Noviomagus
Neophytos, 87, 88, 129, 295
Nesselmann, G. H. F., 31, 41, 60, 101,
235
Netto, E., 211
Newton, John, 249, 305
Newton, Sir Isaac, 196, 252, 253, 386;
aggregation, 345; decimals, 285, 286;
equality, 266, 267; exponential no-
tation, 294, 303, 304, 307, 308,
377; radical sign, 331, 333; ratio and
proportion, 253; annotations of
Euclid, 192
Nichols, E. H., arcs, 370
Nichols, F., 287
Nicole, F., 255, 258, 268; parentheses,
352, 354
Nieuwentijt, B., 264, 266
Nipsus, Junius, 322
Nixon, R. C. J.: angle, 361; paral-
lelogram, 365
Nonius. See Nunez
Nordenskiold, E., 64
Norman, Robert, 162
Norton, R., 186, 276
Norwood, 4, 261; aggregation, 351
Notation, on its importance: Oughtred,
187; Waliis, 199; L. L. Jackson, 199;
Tropfke, 199; Treutlein, 199; Bab-
bage, 386
Noviomagus (Bronkhorst, Jan), 97
Numbers, absolute, signs for, Hindus,
107
Numerals: alphabetic, 28, 29-31;
Arabic (early), 45; Arabic (later), 86;
Aztec, 66; Babylonian, 1-15; Brah-
mi, 77; Chinese, 69-73; Cretan, 32;
Egyptian, 16-25; Fanciful hypoth-
eses, 96; forms of, 85, 86; freak forms,
89; Gobar numerals, 86; Greek, 32-
44, 87; grouping of, 91-94; Hindu-
Arabic, 74, 127; Kharoshthi, 77;
Phoenicians and Svrians, 27, 28;
relative size, 95; Roman, 46, 47;
Tamul, 88; North American Indians,
67; Peru, 62-64; negative, 90
Nufiez, Pedro: survey of his signs. 166,
204; aggregation, 343; several un-
knowns, 161
Nunn, T. P., quoted, 311
Ocreatus, N., 82
Octonary scale, 67
Ohm, Martin, 312; principal values of
a», 312
Oldenburgh, H., 262. 308, 344, 377;
aggregation, 344, 345
ALPHABETICAL mDEX
445
Oliver, Wait, and Jones (joint authors).
210, 213
Olleris, A., 61,
Olney, E., 287
Omicron-sigma, for involution, 307
Oppert, J., 5
Oresme, N., survey of his signs, 123;
129, 308, 333
Ottoni, C. B., 258; angle, 361
Oughtred, William: survey of his signs,
180-87; 91, 148, 169, 192, 196, 205,
210, 218, 231, 236, 244, 248, 382, 385;
aggregation, 343, 345, 347-^9; arith-
metical proportion, 249 j cross for
multiplication, 285; decimals, 283;
equality, 261, 266; geometrical pro-
portion, 251-53, 255, 256; greater or
less, 183; pictographs, 359; powers,
291; radical signs, 329, 332, 334;
unknown quantity, 339
Ozanam, J., 257, 264; equality, 264,
265, 266, 277; powers, 301, 304;
radical sign, 328
Pacioli, Luca: survey of his signs, 134-
38; 91, 117, 126, 132, 145, 166, 177,
200, 219, 220, 221, 222, 223, 225, 226,
294, 297, 359, 384; aggregation, 343;
equality, 138, 260; powers, 297, 322;
radix, 292, 297, 318, 199; unknown,
339
Fade, IL, 213
Palmer, C. I., and D. P. Taylor, equal
number of degrees, 363
Panchaud, B., 249, 259
Paolo of Pisa, 91
Pappus, 55; circle, 367; pictographs,
357
Parallel lines, 359, 368
Parallelogram, pictograph for, 357,
359, 365
Pardies, G., 206, 253, 255
Parent, Antoine, 254, 255, 258; equal-
ity, 263; unknowns, 341
Parentheses: survey of, 342-52; braces,
188, 351; brackets. 347. 351; round,
in Clavius, 161; Girard, 164; He*ri-
gonej 189; Leibniz, 197, 238; mark-
ing index of root, 329; Oughtred,
181, 186. See Aggregation
Paricius, G. H., 208, 262
Parker, 331
Pascal, B., 261, 304, 307; lettering fig-
ures, 376
Pasquier, L. Gustave du, 269
Pastor, Julio Key, 165, 204
Paz, P., 274
Peano, G., 214, 275, 288; aggregation,
348; angle, 362; principal values of
roots, 337; "sgn," 211; use of w, 372
Peet, T. E., 23, 200, 217
Peirce, B., 247, 259, 287; algebraic
symbols, 371
Peise, 14
Peletier, Jacques: survey of his signs,
172; 174, 204, 227, 292; aggregation
in radicals, 332
Pell, John, 194, 237, 307, 386
Pellizzati. See Pellos, Fr.
Pellos, Fr., 278
Penny, sign for, 275
Per cent, 274
Pereira, J. F., 258
Perini, L., 245
Perkins, G. R., 287
Perny, Paul, 69
Perpendicular, sign for, 359, 364
Peruvian knots, 62-64, 69; Peru MSS,
92
Peruzzi, house of, 54
Peurbach, G., 91, 125
Phillips, A. W., and Irving Fisher:
equivalent, 375; spherical excess,
380
Phoenicians, 27, 36
Pi (x): for "proportional," 245; - and
D, 196 '
Picard, J., 254
Piccard, 96
Pictographs, 357-71, 384, 385
Pihan, A. P., 25, 30, 73
Pike, Nicolas, 91, 289
Pires, F. M., 258
Pitiscus, B., 279-81
Pitot, H., 255, 341
Planudes, Maximus, survey of his
signs, 121; 87
Plato, 7
Plato of Tivoli, 290, 322; arcs of circles,
359, 370
Playfair, John: angle, 360; algebraic
symbols, 371; edition of Euefid, 386
Pliny, 50
Plucker, J., 387
"Plus or minus," 210, 196; Leibniz,
198; Descartes, 262, 210
446
A HISTORY OF MATHEMATICAL NOTATIONS
Plus signs: general survey of, 201-16;
186, 199, in Bakhshall, 109; Bom-
belli, 144; Cavalieri, 179; Cardan,
140; Clavius, 161; letter e, 139;
not used in early arithmetics, 158,
Rccorde, 167; Scheubel, 158; shapes
of, 265; sign p, 131, 132, 134, 139,
142, 172-74, 200; spread of + and -,
204, Tartaglia, 143, Vieta, 176,
Widmann, 146
Poebel, Arno, 15
Poinsot, L., 314
Polemi, G., aggregation, 345, 353
Polynier, P., 266
Porfirio da Motta Pegado, L., 248, 258
Pott, A. F., 66
Potts, Robert, 289
Pound, sign for, 275
Powers: survey of, 290-315; Arabic
signs, 116; Bombelli, 144; Cardan,
140; complicated exponents, 313;
Digges, 170; expressed by V, 331;
fractional, 123, 129; in geometry,
307; Ghah'gai, 139; general remarks,
315; Girard, 164; Grammateus, 147;
Hindu signs, 107, 110, 112; Hume,
190; irrational, 308; negative and
literal, 131, 195, 308, 311; Nufiez,
165; Pacioli, 134. 135; principal
values, 312; Peletier, 172; Psellus,
117; repetition of factors, 305;
Recorde, 167; Rudolff, 148; square in
Egyptian papyrus, 100; Schoner,
322; Stifel, 151; Tartaglia, 142, 143;
variable exponents, 310; Vieta, 176,
177; Van der Hoecke, 148; Wallis,
291; fifth and seventh, 135; aa for
a2, 304
Powers: additive principle in marking,
101, 111, 112, 117, 124; multiplica-
tive principle in marking, 101, 111,
116, 135, 142
Powers, S., 65
Praalder, L., 208, 336
Prandei, J. G., 336
Prestel, M. A. F., similar, 372
Prestet, J., 255, 264; aggregation, 344;
decimals, 283; equality, 266; use of
star, 356
Preston, J., 219
Principal values, 211, 312, 337
Principle of local value. See Local value
Pringsheim, Alfred, limit, 373
Priscian, 53
Progression. See Arithmetical progres-
sion, Geometrical progression
Proportion: survey of, 248-59; al-
Qalasadt, 124; arithmetical propor-
tion, 186, 249, 255; continued pro-
portion, 254; compound proportion,
218, 220; geometrical proportion,
244, 249, 250, 254-58; Grammateus,
147; in earliest printed arithmetics,
128; Oughtred, 181; proportion in-
volving fractions, 221; Recorde, 166;
Tartaglia, 142; Wallis, 196; varia-
tion, 259
Pryde, James, 289
Psellus, Michael, survey of his signs,
Ptolemy, 41, 43, 44, 87, 125, 218
Puissant, 249
Purbach, G., 91, 125
Purser, W., 186
Quadratic equations, 26
Quaternary scale, 67
Quinary scale, 67
Quibell, J. E., 16
Quipu of Peru and North America,
62-65
Radicals: Leibniz, 198; Wallis, 196;
reduced to same order, 218, 227;
radical sign j/, survey of, 199, 324-
38; radical >/> with literal index,
330, 331
Radix, 290, 291, 292; R for x, 296, 307,
318; R for root, survey of, 318-21;
R for powers, in Pacioli, 136. See
Roots
Raether, 96
Rahn, J. H.: survey of his signs, 194;
205. 208, 232, 237, 385, 386; Archi-
meaian spiral, 307; equality, 266;
powers, 304, 307; radical signs, 328,
333; unknowns, 341; * for multiplica-
tion, 194
Ralphson. See Raphson
Rama-Crishna Deva, 91
Ramus, P.. 164, 177, 204, 290, 291;
lettering figures, 376; use of /, 322
Raphson, J., 210, 252. 285, 305; ag-
gregation, 345; use of a star, 356
Ratdolt, Erhard, 385
Rath, E., 272
Ratio: arithmetical, 245; "composition
of ratios," 216; geometric (survey of),
244, 252; H&igone, 189; Oughtred,
181, 186, 251) 252; not a division,
245, 246; of infinite products, 196;
sporadic signs, 245, 246
ALPHABETICAL INDEX
447
Rawlinson, H., 5
Rawlinson, Rich., survey of his signs,
193
Rawlyns, R., 283
Reaumur, R. A. F. de, 255
Recio, M., 92
Recorde, R. : survey of his signs, 167-
68; 145, 204, 205, 219, 221, 222, 225,
229, 256, 274; equality, 260^70;
plus and minus, 199; radical sign,
327, 328
Rectangle: "mem," 365; pictograph
for, 357, 359, 368
Rees's Cyclopaedia, 363
Regiomontanus: survey of his signs,
125-27; 134, 138, 176, 208, 250;
decimal fractions, 278, 280; equality,
126, 260, 261; lettering figures, 376;
R for "radix," 318; unknown, 339
Regius, Hudalrich, 225, 229
Regula falsi. See False positions
Reinhold, C. L., geometric congruence,
372
Renaldini, C., 206, 307; aggregation,
351
Res ("thing"), 134, 290, 293
Reye, Theodor: angle, 360; use of
different letters, 379
Reyher, S., 262, 263; angle, 361; arc of
circle, 370; circle, 367; geometry,
387; parallel, 368; right angle, 363;
trapezoid, 371
Reymers, Nicolaus, 208, 296
Reyneau, Ch., 266, 308; use of a star,
356
Rhabdas, Nicolas, 42
Rhind papyrus. See Ahmes papyrus
Riccati, Vincente, 258
Ricci, M. A., 250, 263, 301
Richman, J. B., 92
Riemann, G. F. B., s=,374
Riese, Adam, 59, 148, 176, 208; radicals,
326
Rigaud, S. P., 199, 231, 196, 365
Robbins, E. R.: angle, 360; right angle,
363; parallelogram, 365
Robert of Chester, 385. See Karpinski
Robertson, John, 289
Roberval, G. P., 264
Robins, Benjamin, 307
Robinson, H. N., angle, 362
Roby, H. J., 46
Rocha, Antich, 320, 294
Roche, De la. See De la Roche
Roder, Christian, 126
Rodet, L., 201
Rolle, Michael, 82, 206, 255; equality,
264, 304; aggregation, 344; radical
sign, 331; use of #, 321
Roman numerals, 46-61, 92, 93
Romanus, A., 206, 207; aggregation,
343; powers, 296, 297; radical signs,
329, 330, 199; use of g, 320
Ronayne, Philip, 215, 307; "mem" for
rectangle, 365
Roomen, Adriaen van. See Romanus
Roots: survey of, 316-38; al-Qalasadi,
124; Hindus, 107, 108; Leonardo of
Pisa, 122; Nunez, 165; principal
values, 337; Recorde, 168; spread of
I/ symbol, 327; sign ;/ in Rudolff,
148, 155, in Stifel. 153, 155, in
Scheubel, 159, in Stevin, 163, in
Girard, 164, in Peletier, 172, in
Vieta, 177, in Herigone, 189, in
Descartes, 191, V bino, 163; sign #,
survey of, 318-21 ; in Regiomontanus,
126, in Chuquct, 130, 131, in De la
Roche, 132, in Pacioli, 135, in Tartag-
lia, 142, in Cardan, 141, in Bombelli,
144, in Bolognetti, 145, in Scheubel,
159, in Van der Hoecke, 150;
radix relata, 135, 142; Ra. col. in
Scheubel, 159; #/., 135, 141, 165;
radix distincla, 141; radix legata, 144;
# for x, 137, 160, 318; # to mark
powers, 136; $ to mark both power
and root in same passage in Pacioli,
137; L as radical in Gosselin, 175; \/
and dot for square of binomial, 189
Rosen, F., 115
Rosenberg, Karl, 288
Roth, Peter, 208
Rudolff, Chr. : survey of his signs, 148,
149; 168, 177, 203, 204, 205, 221, 222,
225, 227; aggregation, 148; Coss of
1525, 151, 153, 728; Stifel's edition,
155; decimal fractions, 278, 279;
freak numerals, 89, 91, 158; geo-
metrical proportion, 250; radical
sign, 165, 199, 326, 328; unknown
quantity, 339
Ruska, Julius, 45, 83, 97, 290
Ryland, J., 16
Sacrobosco, J. de, 82, 91, 127
Saez, Liciniano, 52, 92
St. Andrew's cross, 218; in complex
numbers, 247
448
A HISTORY OF MATHEMATICAL NOTATIONS
St. Vincent, Gregory, 261; lettering
figures, 376
Salazar, Juan de Dios, 239, 286
Salignacus, B., 291, 322
Sanders, Alan, is measured by, 373
Sanders, W., 252
Sarjeant, Th., 287
Saulmon, 258
Sault, Richard, 248, 329
Saunderson, Nicholas, aggregation, 345
Saurin, Abbe*, 255
SavSrien, A., 240. 248, 249, 259;
angle, 362; circle, 367; omicron-
sigma, 307; perpendicular, 364; pic-
tographs, 365; right angle, 363;
solids, 371
Scales: quinary, in Egypt, 16; duo-
decimal, in Babylonia, 3, in Egypt,
16; vigesimal, in Egypt, 16, Maya,
68; sexagesimal, in Babylonia, 5, 8,
in Egypt, 16. See Decimal scale
Schack, H., 26
Schafheitlin, P., 308, 344, 366
Scherffer, C., 258
Scheubel, Johann: survey of his signs,
158, 159; 160, 174, 176, 204, 319;
aggregation, 343; plus and minus,
199, 768; radical sign, 326, 327; use
of 5, 319
Schey, W., 208
Schlesser, C., 208
Schmeisser, F., 208, 212, 246
Schmid, K. A., 91
Schnuse, C. H., 348
Schone, H., 103
Schoner, Joh., 272
Schoner, L., 291; unknown, 339; use of
J, 322
Schooten, Van. See Van Schooten
Schott, G., 219; powers, 301
Schott, K., 91, 292
Schreiber, Heinrich. See Grammateus
Schrekenfuchs, 0., 218, 221, 222
Schroder, E., 247
Schron, L., 95
Schroter, Heinrich, angle, 362
Schubert, H., complicated exponents,
313
Schur, F., angle, 362
Schwab, J. C., 268
Schwarz, H. A., 356
Schwenter, D., 250
Scott, Charlotte A., angle, 362
Scratch method of multiplication and
division, 128, 133, 195, 241
Sebokht, S., 79
Segner, J. A. de, 91; aggregation, 354;
geometry, 387; imaginary y— 1,
346
S£jour, Du, imaginary >/( — 1), 346
Selling, E., 90
Selmer, E. W., 208
Senes, D. de, 258
Senillosa, F., 239, 248
Senkereh, tablets of, 5
Serra y Oliveres, A., 275; angles, 363
Sethe, Kurt, 16, 17, 18, 21, 22
Sexagesimal system: in Babylonia, 5,
78; Egypt, 16; Greece, 43, 87;
Western Europe, 44; Wallis, 196;
sexagesimal fractions, 12; degrees,
minutes, and seconds, 55, 126
Sfortunati, G., 219, 221, 223
Sgn, 211
Shai, Arabic for "thing," 290
Shelley, George, 253
Sheppard, W. F., parentheses, 355
Sherman, C. P., 96
Sherwin, H., 285
Sherwin, Thomas, 287
Shutts, G. C., congruent in geometry,
372
Sieur de Var. Lezard, L L., 262
Sign oo, 53, 196
Sign -O, 375
Sign co or co, 41, 372, 373
Sign s, 374
Sigiienza, y G6ngora, 51
Silberstein, L., 215
Similar, survey of signs, 372-74
Simon, Max, similar and congruent,
372
Simpson, Th., 259; aggregation, 345;
use of a star, 356
Simson, Robert, Euclid, 372, 386
Slack, Mrs., 287
Slau^ht, H. E., and Lennes, N. J.
(joint authors), 213; congruent in
geometry, 372
Slusius, R. F., 254, 263; equality, 263
Smith, C., angle, 360
Smith, D. E., 47, 74, 80, 81, 147, 154
208, 274, 278. See also Beman and
Smith
ALPHABETICAL INDEX
449
Smith, Eugene R.: arcs, 370; congru-
ent in geometry, 372; s, 374
Smith, George, 5
Snell; W.? 219; aggregation, 348;
radical signs, 332
Solidus, 275, 313
Solomon's ring, 96
Spain, calderdn in MSS, 92, 93
Speier, Jacob von, 126, 318
Spenlin, Gall, 208, 219
Spherical excess, 380
Spielmann, I., 288
Spier, L., 65
Spitz, C., 213; angle, 362; similar, 372
Spole, Andreas, 265, 301
Square: Babylonians, 15; D to mark
cubes, in Chuquet, 131; to mark cube
roots, in De la Roche, 132; d for
given number, in Wallis, 196;
pictograph, 357, 359, 365; as an
operator, 366
Square root: Babylonian, 15; al-
Qalasadt's sign, 124; Egyptian sign,
100
gridhara, 112; fractions, 271
Staigmuller, H., 159
Stampioen, J., 250, 256. 259; aggrega-
tion, 347, 351; equality, 266; expo-
nents, 299, 303, 307; radicals, 329;
radical signs, 333, 335; D as an
operator, 366
Star: to mark absence of terms, 356;
for multiplication, 194? 195, 232;
in Babylonian angular division, 358
Steele, Robert, 82, 274
Steenstra, P.: angle, 360; right angle,
363; circle, 367
Stcgall, J. E. A., 323
Stegman, J., 283
Stcincr, Jacob: similar, 372; and
Pluckcr, 387
Steinhauser, A., 288
Steinmetz, M., 221, 223
Steinmeyer, P., 219
Sterner, M., 96, 262
Stevin, S.: survey of his signs, 162,
163; 123, 164, 190, 217, 236. 254,
728; aggregation, 343; decimal frac-
tions, 276, 282, 283; powers, 296,
308; lettering figures, 377; radicals,
199, 329, 330, 333; unknowns, 339,
340
Steyn, G. van, 208
Stifel, M.: survey of his signs, 151-56;
59, 148, 158, 161, 167, 169, 170, 171,
172, 175, 176, 177, 192, 205, 217,
224, 227, 229, 236, 384; aggrega-
tion, 344, 348, 349, 351; geometric
proportion, 250; multiplication of
fractions, 152; repetition of factors,
305; radical sign, 199, 325, 327, 328,
329, 334; unknowns, 339
Stirling, James, 233, 354
Stokes, G. G., 275
Stolz, O., and Gmeincr, J. A. (joint au-
thors), 213, 214, 268; angle, 361;
principal values, 312, 337; solidus,
275; uniformly similar, 373
Stone, E., angle, 360, 363
Streete, Th., 251
Stringham, I., multiplied or divided
by, 231
Study, E., 247
Sturm, Christoph, 257
Subtraction, principle of: in al-
Qalasadf, 124; in Babylonia, 10; in
India, 49; in Rome, 48, 49
Subtraction: survey of, 200-216;
Diophantus, 103; Hindus, 106, 108,
109, 114, 200; Greek papyri, 200
Sun-Tsu, 72
Supplantschitsch, R., 288
Surd, sign for, Hindu, 107, 108
Suter, H., 81, 235, 271, 339
Suzanne, H., 248
Swedenborg, Em., 206, 207, 245, 248,
258; equality, 268
Symbolism, on the use of, 39, 40; by
Stifel, 152. See Sign
Symbolists versus rhetoricians, 385
Symbols: value of, 118; by Oughtred,
187
Symmetrically similar triangles, 373
Symmetry, symbol for, 371
Syncopated notations, 105
Syrians, 28, 36
Tacquet, A., 261, 269, 283, 307; alge-
braic symbols, 371
Tamul numerals, 88
Tannery, P., 42, 88, 101, 103, 104, 117,
121, 217, 235, 254, 300, 344, 357
Tartaglia, N.: survey of his signs, 142,
143; 145, 166, 177; 219, 221, 222, 225,
229, 384; geometrical proportion, 250,
254; parentheses, 351; D as an oper-
ator, 366; use of fi, 199, 319
450
A HISTORY OF MATHEMATICAL NOTATIONS
Ternary scale, 67
Terquem, 132
Terrier, Paul, 379
Texeda, Gaspard de, 92
Theon of Alexandria, 87
Thierfeldern, C., 208
Thing. See Cosa
Thompson, Herbert, 42
Thomson, James, 241
Thornycroft, E., 354
Thousands, Spanish and Portuguese
signs for, 92, 93, 94
Thureau-Dangin, Frangois, 12
Todhunter, I., 286, 287; algebraic
symbols, 715; edition of Euclid, 386
Tonstall, C., 91, 219, 221, 222
Torija, Manuel Torres, 286
Toro, A. de la Rosa, 286
Torporley, N., 305
Torricelli, E., 261
Touraeff, B., 100
Transfinite ordinal number, 234
Trenchant, J., 219, 221
Treutlein, P., 96, 147, 148, 151, 154,
156, 263, 296, 340; quoted, 199
Treviso arithmetic, 86, 221
Triangles, pictograph for, 357, 359,
365; sas and as a, 381
Tropfke, J., 91, 136, 140, 149, 151, 159,
176, 201, 203, 217, 255, 263, 277,
289, 293, 296, 324, 340, 343, 344,
348, 353, 359, 376, 386; quoted, 199
Tschirnhaus, E. W. von, 266
Tweedie, Ch., 354
Twysden, John, 186
linger, F., 81, 91, 208
Ungnad, A., 14
Unicorno, J., 226
Unknown number: survey of, 399-41;
Ahmes papyrus, 16; al-Qalasadf, 124;
Bakhshall MS, 109; Catakli, 340;
Chinese, 120; Digges, 170; Diophan-
tus, 101; Hindus, 107, 108, 112,
114; Leibniz, 198; more than one
unknown, 136, 138, 140, 148, 152,
161, 173. 175, 217, 339; Pacioli,
134; Pseilus, 117; Kegiomontanus,
126; represented by vowels, 164,
176; Roman numerals in Hume, 190;
Oughtred, 182, 186;Rudolff, 148, 149,
151; Schoner, 322; Stevin, 162, 217;
Stifel, 151, 152; Vieta, 176-78;
fifth power of, 117
Valdes, M. A., 275, 372
Vallin, A. F., 286
Van Ceulen. See Ludolph van Ceulen
Van Dam, Jan, 52, 208
Van der Hovcke, Daniel, 208
Van der Hoecke, Gielis, survey of his
signs, 150; 147, 204, 319
Vandermonde, C. A., imaginary i/~l,
346
Van der Schuere, Jacob, 208
Van Musschenbroek, P., 267
Van Schooten, Fr., Jr., 176, 177, 210,
232; aggregation, 344, 351; decimals,
283; difference, 262; equality, 264;
geometrical proportion, 254; letter-
ing of figures, 377; pictographs, 365;
powers, 296, 304, 307, 308; radical
sign, 327, 329, 333; use of a star, 356;
comma for multiplication, 232
Van Steyn, G., 208
Variation, 259
Varignon, P., 255; radical sign, 331;
use of star, 356
Vaulezard, J. L. de, 351
Vazquez, M., 258, 286
Vectors, 373
Venema, P., 259; radical signs, 336
Veronese, G. : congruent in geometry,
374; not equal, 383
Vieta, Francis: survey of his signs,
176-78; 188, 196, 204, 206, 262, 384;
aggregation, 343, 344, 351. 353;
decimal fractions, 278; general expo-
nents, 308; indicating multiplica-
tion, 217; I for lotus, 290, 322, 327;
letters for coefficients, 199, 360;
powers, 297, 307; radical sign y, 327,
333; use of vowels for unknowns,
164, 176, 339
Vigesimal scale: Aztecs, 66; Maya, 68;
North American Indians, 67
Villareal, F., 286
Vinculum: survey of, 342-46; in
Bombelli, 145; Chuquet, 130; H6ri-
gone, 189; joined to radical by Des-
cartes, 191, 333; Leibniz, 197;
Vieta, 177
Visconti, A. M., 145, 199
Vitalis, H., 206, 268, 269; use of fi,
321
Vnicorno, J. See Unicorno, J.
Voizot, P., 96
Voss, A., 211
ALPHABETICAL INDEX
451
Wachter, G., 96
Waessenaer, 297
Walkingame, 331
Wallis, John: survey of his signs. 195,
196; 28, 132, 139, 186, 192, 196,
210, 231, 237, 248, 264, 384, 385,
386; aggregation, 345, 347, 348,
353; arithmetical proportion, 249;
equality, 266; general exponents,
308, 311; general root indices, 330;
geometrical proportion, 251, 252;
imaginaries, 346; lettering figures,
376; parentheses, 350; quoted, 195,
199, 307; radical signs, 330, 332;
sexagesimals, 44
Walter, Thomas, 330
Walther, J. L., 48, 201
Wappler, E., 50, 82, 201, 324
Ward, John, 252, 307, 386; angle, 361;
circle, 367
Ward, Seth, 251
Waring, E.: aggregation, 351; compli-
cate^ exponents, 313; imaginary
/-I, 346; use of a star, 356
Weatherburn, C. E., 215
Webber, S., 287
Webster, Noah, 340
Weidler, 96
Weierstrass, K., use of star, 356
Weigel, E., 207, 249, 255; equality, 266,
268
Wells, E., 231, 252, 285
Wells, Webster: arcs, 370; equivalent,
375
Wells, W., and W. W. Hart: congruent
in geometry, 372; s=, 374
Wentworth, G. A.: angle, 360; equiva-
lent, 375; parallelogram, 365; right
angle, 363
Wertheim, G., 148, 173, 296, 339
Wersellow, Otto, 208
Whiston, W., 248, 269, 285, 286, 307;
edition of Tacquet's Euclid, 386;
radical signs, 331 ; use of a star, 356
White, E. E., coincides with, 374
Whitworth, W. A., dot for multiplica-
tion, 233
Widman, Johann, 146, 201, 202, 205,
208, 219, 221, 272, 384; radical
sign, 293, 318
Wieleitner, H., 264, 326, 341
Wilczynski, E. J., 524
Wildermuth, 91
Wilkens, 212
Wilson, John, 248, 253; angle, 362;
perpendicular, 364; parallel, 368
Wing, V., 44, 244, 251, 252, 253, 258
Wingate, E., 222, 251, 283; radical
signs, 332
Winterfeld, von, 212, 246
Witting, A., 159
Woepcke, F., 74, 118, 124, 339
Wolf, R., 278
Wolff (also Wolf), Chr., 207, 233, 238;
arithmetical proportion, 249; astro-
nomical signs, 358; dot for multipli-
cation, 285; equality, 265; geo-
metrical proportion, 255, 259; radi-
cal sign, 331; similar, 372
Wolletz, K., 288
Workman, Benjamin, 225, 254
Worpitzky, J., notation for equal tri-
angles, 381
Wren, Chr., 252
Wright, Edward, 231, 261; decimal
fractions, 281
Wright, J. M. F., quoted, 386
Xylander (Wilhelm Holzmann), 101,
121, 263, 178; unknowns, 339
York, Thomas, 225, 245, 254
Young, J. W., and A. J. Schwartz,
congruent in geometry, 374
Ypey, Nicolaas, 257
Zahradnicek, K., 288
Zaragoza, J., 219, 254; radical signs,
330
Zero: symbol for, 5, 11, 68, 84, 109;
forms of, in Hindu- Arabic numerals,
81, 82, 83; ornicron, 87
Zirkel, E., 218
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